Structural Steelwork Eurocodes 1
Introduction to EC4
Eurocode 4 Part 1.1 2
Sections
– – – –
follow a typical design sequence:
Material properties Safety factors Methods of analysis Element design, ultimate and serviceability
Some
sections deal with specific topics:-
– Durability – Composite joints in frames for buildings – Composite slabs with profiled steel sheeting
Terminology 3
A number of terms are clearly defined: – „COMPOSITE MEMBER‟ - one with concrete and steel interconnected components • „BEAM‟ - subject mainly to bending • „COLUMN‟ - subject mainly to compression • „SLAB‟ - profiled steel sheets as permanent shuttering and tensile reinforcement – „SHEAR CONNECTION‟ - the connection between steel and concrete components
Terminology 4
„COMPOSITE FRAME‟ - includes some composite elements „COMPOSITE JOINT‟ - reinforcement contributes to resistance ad stiffness. „PROPPED STRUCTURE OR MEMBER‟ - weight of wet concrete carried independently, or steel supported until concrete able to resist stress „UNPROPPED STRUCTURE OR MEMBER‟ - weight of wet concrete is applied to the steel elements unsupported in the span.
Notation/Symbols 5
Symbols – – – – – – – –
L,1 N R S
of a general nature:
Length; span; system length Number of shear connectors; axial force Resistance; reaction Internal forces & moments; stiffness Deflection; steel contribution ratio Slenderness ratio Reduction factor for buckling Partial safety factor
Notation/Symbols 6
Symbols
– – – – – – – –
A b d h i I W
relating to section properties:
Area Width Depth; diameter Height Radius of gyration Second moment of area Section modules Diameter of a reinforcing bar
Notation/Symbols 7
Symbols –E –f –n
relating to material properties:
Modules of elasticity Strength Modular ratio
Notation/Symbols 8
Subscripts: –c Compression, composite cross-section, concrete –d Design – el Elastic –k Characteristic – LT Lateral-torsional – pl Plastic
Material Properties 9
Concrete – Normal and lightweight concrete as EC2 – Concrete grades less than C20/25 or greater than C60/75 excluded Reinforcing
steel
– As EC2 – Reinforcement grades with a characteristic strength greater than 550N/mm2 not covered
Material Properties 10
Structural
steel
– As EC3 – Steel grades with a characteristic strength greater than 460N/mm2 not covered Profiled
steel sheeting for composite slabs
– As EC3 – Types of steel restricted – Recommended minimum thickness is 0.7mm Shear
connectors
– Reference to ENs for material specification
Structural Analysis 11
Ultimate
Limit State
– Elastic or plastic global analysis allowed – Certain conditions apply to the use of plastic analysis Serviceability
Limit State
– Elastic analysis must be used – The effective width is as defined for the ultimate limit state, and appropriate allowances may be made for concrete cracking, creep and shrinkage
Elastic Analysis 12
Stages of construction may need to be considered The stiffness of the concrete may be based on the uncracked condition for braced structure In other cases, some account may need to be taken of concrete cracking by using a reduced stiffness over a designated length of beam Creep is accounted for by using appropriate values for the modular ratio Shrinkage and temperature effects may be ignored Some redistribution of elastic bending moments is allowed
Rigid-Plastic Global Analysis 13
Allowed
for
– non-sway frames – unbraced frames of two storeys or less Some
restrictions on cross sections
Properties and Classification of Cross-Sections 14
The
effective width of the concrete flange is defined More rigorous methods of analysis are admitted Cross-sections are classified in a similar manner to EC3 for non-composite steel sections
Ultimate Limit State 15
Concerned
with the resistance of the structure to collapse Based on the strength of individual elements Overall stability of the structure must be checked Factored load conditions
Beams 16
Bending
resistance
– Applicability of plastic, non-linear and elastic analysis – Full or partial interaction defined Vertical
shear resistance
– Effects of shear buckling – Combined bending and shear
Partially Encased Beams 17
ď ľ Concrete
infill between the flanges enclosing the web ď ľ Separate considerations apply for bending and shear
Lateral-Torsional Buckling 18
Top flange is laterally restrained by the
concrete slab In hogging bending the compression flange is not restrained – Lateral-torsional buckling must be checked – Under certain conditions such checks are unnecessary
Longitudinal Shear Connection 19
Related
to:
– strength of slab and transverse reinforcement – connector types
Columns 20
Various
types of composite columns
– Encased sections – Concrete-filled tubes Simplified
procedures for doubly symmetric cross-sections Guidance is given on shear connection
Serviceability Limit State 21
Deflections Concrete
cracking Control of vibrations and limiting stresses are not included
Deflections 22
Calculated
deflection is seldom meaningful because: – actual load unlike design load; – idealised support conditions seldom realised
But
calculated deflection can provide an index of stiffness
Deflections 23
Guidance
is given on calculating deflections for composite beams – including allowances for partial interaction – concrete cracking
No
guidance is given regarding simplified approaches based on limiting span/depth ratios.
Deflection Limits 24
ď ľ No
reference to limiting values for deflections ď ľ Calculated deflections should be compared with limits in Eurocode 3
Deflection Limits 25
Six
categories:
– roofs generally – roofs frequently carrying personnel other than for maintenance – floors generally – floors and roofs supporting plaster or other brittle finish or non-flexible partitions – floors supporting columns (unless deflection included in global analysis for ultimate limit state) – situations in which the deflection can impair the appearance of the building
Calculating Deflections 26
Steel
member alone
– Construction stage for for unpropped conditions – Procedures of EC3 – Bare steel section properties Composite
cross-section
– Elastic analysis – Suitable transformed section – Allow for incomplete interaction and cracking of concrete where appropriate
Concrete Cracking 27
Concrete
may crack due to:
– Direct loading – Shrinkage Excessive
cracking of the concrete can:
– affect durability – compromise appearance – impair the proper functioning of the building
Concrete Cracking 28
May
not be critical issues Simplified approaches based on: – minimum reinforcement ratios – maximum bar spacing – diameters Guidance
on calculating crack widths Limiting crack widths related to exposure conditions
Composite Joints 29
Moment-resisting
connections Calculations for: – moment resistance – rotational stiffness – rotation capacity
beam-column
Composite Joints 30
Inter-dependence of global analysis and connection design – may be neglected where the effects are small
Classification - stiffness rigid nominally pinned semi-rigid
Classification - strength full strength nominally pinned partial strength
Guidance on design and detailing of the joint, including slab reinforcement
Composite Slabs 31
Ultimate
and serviceability limit states Construction stage – steel sheeting acts as permanent shuttering – must support wet concrete loads (unpropped) – reference to EC3 Part 1.3
Composite Slabs 32
Calculation – – – – – –
procedures for
flexure longitudinal shear vertical shear stiffness span:depth ratios limiting
Concluding Summary 33
A number of terms in EC4 have a very precise meaning The principal components for composite construction are concrete, reinforcing steel, structural steel, profiled steel sheet, and shear connectors Material properties for each component are defined in other Eurocodes Guidance is given on what methods of analysis, both global and cross-sectional, are appropriate
Concluding Summary 34
EC4 is based on limit state design principles The Ultimate Limit State is concerned with collapse The Serviceability Limit State is concerned with operational conditions. These relate specifically to deflections and crack control, and EC4 provides guidance for controlling both EC4 is structured on the basis of element type; detailed procedures for the design of beams, columns and slabs are given in separate sections.
Structural Steelwork Eurocodes 35
Structural Modelling and Design
Scope of the lecture 36
Structural
The
modelling
design process
Generalities
about design requirements for main structural elements
Structural modelling and the design process 37
DEFINITION OF THE STRUCTURE Geometry Load cases and combinations
THE STRUCTURAL MODELLING Structural concept Main structural elements Frame classification
Addressed in the present lecture THE DESIGN PROCESS Structural frame analysis Verification SLS and ULS
Structural modelling 38
Idealisation
of the actual frame and derivation of a structural model to be used in the frame design and analysis process Reference to Annex H of Eurocode 3 Annex H also applicable to composite structures Some aspects briefly addressed here
Structural modelling 39
Structural
concept
A structure is composed of: – Main elements: main frames, their members, joints and foundations; they transfer all the vertical and horizontal forces to the foundations – Secondary elements: purlins, secondary beams; they transfer loads to the main structural elements – Other elements: sheeting, roofing, partitions, … (continued)
Structural modelling 40
Spatial behaviour Reduction of a three-dimensional framework to plane frames
(continued)
Structural modelling 41
Resistance to horizontal forces Frame classification into : – braced/unbraced – sway/nonsway frames introduced later on in the present lecture
(continued)
Structural modelling 42
Ground-structure interaction
The following procedure is proposed in Annex H for examining ground-structure interaction : – As a first step, the structure may be analysed assuming that the ground is rigid. From this analysis, the loading on the ground should be determined and the resulting settlements should be calculated. – The resulting settlements are applied to the structure in the form of imposed deformations and the effects on the internal forces and moments should be evaluated. – When the effects are significant (more than 5%), the groundstructure interaction should be accounted for. This may be done by using equivalent springs to model the soil behaviour. (continued)
Structural modelling 43
Modelling of frames
The following guidelines are taken from Annex H: – The members and joints should be modelled for global analysis in a way that appropriately reflects their expected behaviour under the relevant loading. – The basic geometry of a frame should be represented by the centrelines of the members. – It is normally sufficient to represent the members by linear structural elements located at their centrelines, disregarding the overlapping of the actual widths of the members.
Alternatively, account may be taken of the actual width of all or some of the members at the joints between members. Methods which may be used to achieve this are proposed in Annex H. (continued)
Structural modelling 44
Framing and joints Framing is used to distinguish between the various ways that joint behaviour can be considered for global analysis (possible discontinuities between members). – Continuous framing: the discontinuity may be neglected, i.e. the joints are assumed to be rigid, and the frame may be analysed as continuous. – Simple framing: the discontinuity may be taken into account by assuming a pinned (hinged) joint model, taking advantage of possible rotations without considering joint moment resistances. – The discontinuity at the joints may also be accounted for by using semi-continuous framing. This is where a joint model (i.e. a semi-rigid joint model) is used in which its momentrotation behaviour is taken into account more precisely. (continued)
Structural modelling 45
ď ľ Main
structural elements composite slabs composite beams steel or composite columns
Structural modelling 46
Specifity of composite frames
The reduction of a three-dimensional framework to plane frames is a general principle for modelling This principle, which is easily applied to steel structures, requires further attention as far as composite buildings are concerned. The specific reason for this is the presence at each floor of a two-dimensional composite slab. (continued)
Structural modelling 47
In order to overcome this difficulty: – The slab is assumed to span in a principal direction and is designed accordingly. – The three-dimensional framework is then reduced to plane frames which are studied independently each from the other. – In order to enable such a “dissociation” into plane frames, the concept of effective width for the composite slabs is introduced.
As a result, a composite beam is constituted by a steel profile and an effective slab; both components are connected together by shear connectors. (continued)
Structural modelling 48
Effective width of slab
« Shear lag » effects induce non uniform stress distribution in the slab concept of “effective width” beff b b e1
b1
b1
eff b e2
b2
(continued)
Structural modelling 49
Simple and safe recommendations in Eurocode 4:
beff = be1 + be2 with where
bei = min ( Lo/8; bi ) Lo is the distance measured between consecutive points of contraflexure in the bending moment diagram
L0 =
0,25(L 1+ L2 )
L0 =
0,8L1
L1
0,25(L 2 + L3 )
0,7L2
L2
1,5L4 but < L4+0,5L3 )
0,8L3 - 0,3L4 but > 0,7L3 L3
(continued) L4
Structural modelling 50
Equivalent modular ratio n
Actual composite beam cross-section replaced by an equivalent steel section for elastic calculations. Use of an « equivalent modular ratio n »
Ea n Ec' Ea is the elastic modulus for steel E’c is an elastic « effective »modulus for concrete
(continued)
Structural modelling 51
The transformed section
(continued)
Structural modelling 52
Elastic strains and stresses in the composite section
(continued)
Structural modelling 53
The E’c value is influenced by: – the concrete grade – the concrete age, – the short-term or long-term character of the loading Effects of creep, shrinkage, …
Values of E’c ond of the modular ratio n are given in the presentation of Lecture 5
Structural modelling 54
Frame
classification
Two classification criteria: – Braced / Unbraced
– Sway / Non-sway
(continued)
Structural modelling 55
Braced and unbraced classification criteria
If no bracing system is provided: the frame is unbraced. When a bracing system is provided: – when br > 0,2 unbr : the frame is classified as unbraced – when br 0,2 unbr : the frame is classified as braced
where:
br unbr
is the lateral flexibility of the structure with the bracing system. is the lateral flexibility of the structure without the bracing system. (continued)
Structural modelling 56
Sway and non-sway classification criteria
Non-sway frames – the frame response to in-plane horizontal forces is sufficiently stiff for it to be acceptable to neglect any additional forces or moments arising from horizontal displacements of its nodes – the global second-order effects (i.e. the P- sway effects) may be neglected – a first order structural frame analysis is used
Sway frames – the global second-order effects are not negligible and a second order analysis is required
(continued)
Structural modelling 57
Sway / Non-sway classification criteria – Non-sway frames
VSd 0,1 Vcr
(main scope of EC4)
– Sway frames
VSd 0,1 Vcr (continued)
The design process 58
PREDESIGN OF THE MAIN STRUCTURAL ELEMENTS (Columns, beams, slabs, joints)
STRUCTURAL FRAME ANALYSIS (Distribution of internal forces, displacements)
DESIGN VERIFICATIONS Verification SLS and ULS
Addressed in the SSEDTA lectures
The design process 59
Frame
analysis
Lecture 5
Design
of slabs
Lecture 4
Design
of beams
Lecture 6 and 7
Design
of columns
Lecture 8
Design
of joints
Lecture 9
Design requirements 60
Design
requirements at SLS
– Control of: the transverse displacements of the composite beams the cracking of the concrete the beam vibrations, especially for large span beams (continued)
Design requirements 61
â&#x20AC;&#x201C; Possible simplifications: ď&#x201A;§ The influence of the concrete shrinkage on the transverse beam displacements is only taken into consideration for simply supported beams with a "span-to-total section depth" ratio higher than 20 and providing that the free shrinkage deformation likely to occur exceeds 4 x 10-4. ď&#x201A;§ It is permitted to simplify the elastic analyses of the structural frame through the adoption of a single equivalent modular ratio n" for the concrete which associates the creep deformations under long-term actions and the instantaneous elastic deformations.
Design requirements 62
Design
requirements at ULS
– Verification of the joint resistance – Verification of the slab resistance
– Verification of the column stability and resistance – Verification of the composite beams (specific design checks listed on next slides)
(continued)
Design requirements 63
– Specific design checks for composite beams Resistance of critical sections defined as the points of : maximum bending moment (section I-I) maximum shear (section II-II at external supports) high combined bending moment and shear (sections III-III) sudden change of section and/or mechanical properties P d II
I
III
VI
III IV
V
VI II
I
III
V
(continued)
III splice
V
Design requirements 64
– Specific design checks for composite beams The strength of the longitudinal shear connection (line IV-IV) The longitudinal shear strength of the transversally reinforced concrete slab (line V-V and VI-VI) The resistance to lateral-torsional buckling under negative bending moments, with lateral displacement of the bottom flange of the steel section (buckledPdposition VII)
II
I
III
VI
III
VI
IV VI II
I
III
V
V VI
III splice
VII
Structural Steelwork Eurocodes
Composite Slabs with Profiled Steel Sheeting
Composite slabs • One way spanning
• Typical span 3.5 m • Span onto composite secondary beams
• Secondary beams span onto primary beams • Rectangular grids
• Slab unsupported during construction
Advantages of composite slabs speed and simplicity of construction safe working platform protecting workers below lighter than traditional concrete building often used with lightweight concrete
- reduces the dead load factory manufactured decks and beams
- strict tolerances achieved
Composite slabs comprise of • steel decking • reinforcement
in-situ concrete slab
• cast in situ concrete
Support beam
After concrete has hardened: behaves as a composite steelconcrete structural element
reinforcement
Support beam
After construction:
Profiled steel designed to act as both permanent formwork during • profiled steel sheet concreting and tension • upper concrete topping reinforcement interconnected so that horizontal shear forces can be transferred at the steelconcrete interface.
Profiled decking types Numerous types with different: • shapes • depth and distance between ribs • width, lateral covering, • plane stiffeners • mechanical connections Thickness : 0,75mm 1,5mm Depth : 40mm 80mm Galvanized on both faces Cold formed Cold forming causes strain hardening and increase in yield S235 300 N/mm2
Steel to concrete connection • Adhesion not sufficient to create composite action in the slab
bo
bo hc hp b
b
re-entrant trough profile
Efficient connection made by: Mechanical anchorage from local deformations Decking shape - re-entrant trough profile End anchorage provided by welded studs End anchorage by deformation of the ribs at the end of the sheeting.
( a ) mechanical anchorage
( b ) frictional interlock
hc h
hp b
b
Open trough profile
( c ) end anchorage
( d ) end anchorage by deformation
h
Reinforcement in the slab
Provided for: Load distribution of line or point loads Local reinforcement of slab openings Fire resistance Upper reinforcement in hogging moment area Control cracking due to shrinkage
Mesh reinforcement placed at the top of the profiled decking ribs.
Design requirements Overall depth h, > 80 mm. Thickness hc of concrete over decking > 40mm
If the slab acts compositely with a beam, or is used as a diaphragm total depth h > 90mm concrete thickness hc > 50mm bo
â&#x20AC;˘ The nominal size of aggregate should not exceed the least of: 0,40 hc or bo/3 or 31,5 mm ď&#x201A;ˇ Composite slabs require a minimum bearing of 75mm for steel or concrete and 100mm for other materials.
bo hc hp
b
b
re-entrant trough profile
hc h
hp b
b
Open trough profile
h
Composite slab behaviour Perfect connection between the concrete and steel sheet - complete interaction.
P
hc hp
b
ht
Ls =
Relative longitudinal displacement between steel sheet and adjacent concrete- incomplete interaction.
P
L 4
Ls =
L 4
L
load P
P
P
ď ¤ P
P u : complete interaction
u
P u : partial interaction
P u : no interaction P
f First crack load 0
deflection ď ¤
Three types of behaviour Complete interaction:
load P
P
P
P
• No global slip at the steelconcrete interface exists • Failure can be brittle or ductile Zero interaction: • Global slip at the steelconcrete interface is not limited and there is almost no transfer of shear force.
P u : complete interaction
u
P u : partial interaction
P u : no interaction P
f First crack load 0
deflection
Partial interaction: • Global slip not zero but limited • Shear force transfer partial and the ultimate lies between the ultimate loads of the previous cases. • Failure can be brittle or ductile.
Composite slab stiffness • • •
Represented by the first part of the P- curve Stiffness highest for complete interaction Three types of link between steel and concrete: 1. Physical-chemical link: always low but exists for all profiles 2. Friction link: develops as soon as micro slips appear 3. Mechanical anchorage link: acts after the first slip depends on the steel-concrete interface shape.
Composite slab stiffness After first cracking, frictional and mechanical interaction begin to develop as the first micro-slips occur.
load P
P
P
ď ¤ P
P u : complete interaction
u
P u : partial interaction
P u : no interaction P
f First crack load 0
From 0 to Pf , the physicalchemical phenomena account for most of the initial interaction between the steel and concrete.
deflection ď ¤
Stiffness depends on the effectiveness of the connection type.
Composite slab collapse modes Failure type I applied moment exceeds Mpl.Rd generally the critical mode for moderate to high spans with a high degree of interaction between the steel and concrete. III
I II
Shear span Ls
Composite slab collapse modes Failure type II ultimate load resistance is governed by the steel concrete interface. happens in section II along the shear span Ls.
III
I II
Shear span Ls
Composite slab collapse modes Failure type III applied vertical shear exceeds shear resistance. This is only likely to be critical for deep slabs over short spans and subject to heavy loads III
I II
Shear span Ls
Brittle or ductile failure? • Depends on characteristics of the steel-concrete interface. • Slabs with open trough profiles experience a more brittle behaviour • Slabs with re-entrant trough profiles tend to exhibit more ductile behaviour.
Load P
• Decking producers ameliorate brittle behaviour with various mechanical means – embossments, indentations, dovetails
Ductile behaviour
Brittle behaviour
deflection
Shear connectors between beam and slab influence the failure mode.
Design conditions
Two design conditions should to be considered During construction In service concrete and steel combine to form a steel sheet acts as single composite unit shuttering
Construction condition • Steel deck must resist weight of wet concrete and the construction loads • Deck may be propped temporarily during construction • Preferable if no propping is used • Verification at ULS and SLS should be in accordance with part 1.3 of Eurocode 3 • Effects of embossments or indentations on the design resistances should be considered
Design (construction) at the ULS Weight of concrete and steel deck Construction loads - weight of the operatives and concreting plant „Ponding' effect - increased depth of concrete due to deflection Storage load (if any) (b)
(a)
(c)
(b)
(b)
(a)
3000
(c)
3000
Moment in mid-span
moment over support
(a)
Concentration of construction loads 1,5 kN / m²
(b)
Distributed construction load 0,75 kN / m²
(c)
Self weight
(b)
! Minimum values are not necessarily sufficient for excessive impact or heaping concrete, or pipeline or pumping loads
Deflection Under self weight + the weight of wet concrete,
but excluding construction loads
< L/180 If < 1/10 of the slab depth
- ponding effect may be ignored Allow for ponding by assuming in design - nominal thickness of the concrete is
increased over the whole span by 0,7.
Composite slab design checks Loads to be considered are the following : 1. Self-weight of the slab (profiled sheeting and concrete) 2. Other permanent self-weight loads (not load carrying elements) 3. Reactions due to the removal of the possible propping 4. Live loads 5. Creeping, shrinkage and settlement 6. Climatic actions (temperature, wind...). For typical buildings, temperature variations are generally not considered.
Serviceability limit state
1. Deflections 2. Slip between the concrete slab and the decking at the end of the slab called end slip 3. Concrete cracking
Deflections
Recommended limiting values
• L/250 permanent + variable long duration loads • L/300 variable long duration loads • L/350 if composite slab supports brittle elements
Deflection of the sheeting due to its own weight and the wet concrete need not be included
End slip External spans - end slip can have a significant effect on deflection. For non-ductile behaviour, initial end slip and failure may be coincident Semi-ductile behaviour - end slip increases deflection. End anchorage may sometimes be necessary to prevent end slip at SLS End slip is considered as significant when it is higher than 0,5mm. No account need be taken of the end slip if the limit is reached for a load >1,2 x service load. Slip @ < 1,2 x service load - end anchors should be provided or deflections should be calculated including the effect of the end slip.
Concrete cracking • Crack widths in hogging moment regions of continuous slabs checked in accordance with EC2 • In normal circumstances (not aggressive exposure) - maximum crack width 0,3mm • Crack width > 0,3mm - reinforcement should be added according to usual reinforced concrete rules • Continuous slabs usually designed as a series of simply supported beams - area of anti-crack reinforcement > 0,2 % area of the concrete on top of the steel sheet for unpropped construction - increased to 0,4 % for propped construction
Analysis for internal forces and moments Profiled steel sheeting as shuttering: • Elastic analysis should be used due to the slenderness of the sheeting cross-section • Second moment of area constant - calculated considering the cross-section as fully effective • Simplification is only allowed for global analysis - not for cross-section resistance - deflection checks
Analysis for internal forces and moments Composite slab analysed by: Linear analysis without moment redistribution at internal supports if cracking effects are considered explicitly
Linear analysis with moment redistribution at internal supports (limited to 30 %) without considering concrete cracking effects Rigid-plastic analysis provided that it can be shown that sections where plastic rotations are required have sufficient rotation capacity
Elastic-plastic analysis taking into account non-linear material properties
Analysis for internal forces and moments • Linear methods suitable for SLS and ULS
• Plastic methods should only used at ULS • Continuous slab may be designed as simply supported spans nominal reinforcement should be provided over intermediate supports
Analysis for internal forces and moments Point or line loads parallel to the span of the slab:
• Considered to be distributed over an effective width • Provide transverse reinforcement to ensure distribution of line or point loads over the effective width
• If characteristic imposed loads < 7,5kN or 5,0kN/m² - use nominal transverse reinforcement area > 0,2% - reinforcement provided for other purposes may fulfil all or part of this requirement.
Verification of deck as shuttering at ULS • Construction load case is one of the most critical
• Refer to part 1.3 of Eurocode 3 for verification • Effective section should account for effects of local buckling. • Determine Ieff and Weff •Check bending moment resistance of the section
M Rd f yp
Weff ap
Deflection of decking at SLS Determined with the second moment of area of the effective section The deflection of the decking under uniformly distributed loads (p) acting in the most unfavourable way on the slab L
L
L
k = 1,00 for simply supported decking; k = 0,41 with two equal spans (3 supports); k = 0,52 with three equal spans; k = 0,49 with four equal spans.
L
5 1 4 ď ¤ď&#x20AC;˝k pL 384 EI eff
Composite slab in sagging • Type I failure is due to sagging bending resistance - reached if the steel sheeting yields in tension or if concrete attains its resistance in compression • Supplementary reinforcement taken into account • Material behaviour idealised as rigid plastic • At ULS steel, concrete and reinforcement at design strength • Anti-cracking reinforcement and tension reinforcement for hogging bending neglected when evaluating resistance to sagging bending • Two cases have to be considered according to the position of the plastic neutral axis
PNA above the sheeting
Concrete in tension ignored Tension force in the steel sheeting
Compression force in the concrete
N p Ape N cf
f yp
ap 0,85 f ck b x pl c
0,85 f
Ape f yp x pl
ap 0,85 b f ck c
N cf Xpl dp
z
d
z d p 0.5 x
M ps.Rd N p z f yp x M ps.Rd Ape (d p ) ap 2
c
ck
Np f yp centroidal axis of profiled steel sheeting
ap
PNA in sheeting Ncf = Np Ncf, = resistance of the concrete slab Np = partial tension force in the steel sheeting. z depends on the deck shape calculated by an approximate method Moment = Ncf..z.
Pair of equilibriating forces in the steel profile. Moment is Mpr , reduced plastic moment of deck add to Ncf z.
0,85 f ck
c N cf
hc dp d
ep
h
z
p.n.a. f yp ap
e
Centroidal axis of profiled steel sheeting
p.n.a. : plastic neutral axis de gravité
=
Np
f yp ap c.g. : centre of gravity
+
M pr
Mpr, reduced (resistant) plastic moment of the steel sheeting
M ps.Rd Ncf z M pr N cf
0,85 f ck bhc c
deduced from Mpa, design plastic resistant moment of the effective cross-section of the sheeting as shown M pr M pa
N cf z ht 0,5hc e p (e p e) Ap f yp
N cf M p r 1,25 M p a (1
1,25 1,00
ap Tests envelope curve
0
Na Ap f yp
A p f yp ap
)
Hogging bending resistance Type I failure due to hogging bending resistance • PNA generally in the deck. • Steel sheeting in compression ignored • Concrete in tension neglected. • All tension carried by reinforcing bars N s As f ys / s Ns As x
f ys s
0,85 bc
M ph.Rd
As f ys s
f ck c
Xpl
z N c 0,85 bc x pl
f ck c
Longitudinal shear Type II failure - resistance to longitudinal shear • Evaluate average longitudinal shear resistance u existing on shear span Ls and compare with applied force. • Resistance u depends on the type of sheeting - must be established for all proprietary sheeting - value is a function of the particular arrangements of embossment orientation, surface conditions etc. • Two methods 1. Semi-empirical method called the m-k method - uses the vertical shear force Vt to check the longitudinal shear failure along the shear span Ls. 2. Partial interaction method
m-k method
B
Design relationship for longitudinal shear resistance
Vt b dp A
( N / mm2 )
P
m
P
1 V
Ls
k 0
V t Ls
Ap bLs
Stress dimension term - depends on vertical shear force Vt includes self-weight of slab.
t
m-k line determined with six full-scale slab tests separated into two groups for each steel profile type.
Non-dimensional number and represents the ratio between the area of the sheeting and the longitudinal shear area.
Direct relationship is established with the longitudinal shear load capacity of the sheeting.
m-k method
VL.Rd b.d p (m
Ap bLs
k)
1 VS
k ordinate at the origin k slope of the m-k line VS is a partial safety factor equal to 1,25. Ls depends on the type of loading uniform load applied to the entire span L simply supported beam, Ls =L/4 Where the composite slab is designed as continuous, use an equivalent simple span between points of contraflexure For end spans the full exterior span length should be used in design.
m-k method Longitudinal shear line is only valid between some limits because, depending on the span, the failure mode can be one of the three shown Vt bdp
vertical shear m Longitudinal shear
k
flexural Long
Ls span
short Ap b Ls
Longitudinal shear resistance can be increased by the use of some form of end anchorage, such as studs or local deformations of the sheeting.
Partial connection method, or u method u given by the profile steel
Alternative method for the resistance to longitudinal shear Only for composite slabs with ductile behaviour Based on the value of the design ultimate shear stress u.Rd acting at the steel-concrete interface
manufacturer or by results from standardised tests on composite slabs
Design partial interaction diagram bending resistance MRd of a cross-section at a distance Lx from the nearer support is plotted against Lx.
0.85 fck / c
M
N cf
Rd N c = b.L x . u.Rd f yp / ap
M pl. Rd
f yp f yp
flexure
longitudinal shear
u.Rd
A N cf
M pa
Lx
L sf =
Ncf
b u.Rd
Lx
A
Partial connection method, or u method
No connection (Lx = 0), assumed that the steel sheeting supports the loading resistance moment Mpa
0.85 fck / c M
N cf
Rd N c = b.L x . u.Rd f yp / ap
M pl. Rd
f yp f yp
flexure
longitudinal shear
u.Rd
M pa
Lx
Ncf
b u.Rd
For full interaction Lsf
b.u.Rd
A N cf
L sf =
N cf
For full connection, design resistant moment Mpl.Rd.
N cf min(
0,85 f ckbhc Ap f yp ; ) c ap
Lx
A
Verification procedure 1. Draw resistant moment diagram 2. Draw design bending moment on the same axis system. 3. For any cross-section of the span, the design bending moment MSd cannot be higher than the M Sd M Rd design resistance MRd. M
pl.Rd
M
Rd M Sd for A LA
M pa
M Sd for B M Sd < M Rd L sf
LA
LB
Lx
LB
Vertical shear failure Type III failure - resistance to vertical shear
Critical where steel sheeting has effective embossments (thus preventing Type II failure) Characterised by shearing of the concrete and oblique cracking
Vv.Rd bo d p k1k2 Rd bo : mean width of the concrete ribs Rd : basic shear strength = 0,25 fctk/c fctk : approximately 0,7 x mean tension resistance fctm Ap : effective area of the steel sheet in tension
hc
bo
k1 (1,6 d p ) 1
k2 1,2 40 Ap / bo d p 0,02
dp
Punching resistance
hc
hc
Critical perimeter C p
dp
Loaded area
dp
hc
V p.Rd C p hc k1k2 Rd
Elastic properties for SLS
Compression zone
xc dp
hc
E.N.A.
xu
Compression zone E.N.A.
Steel sheeting centroid axis
Tension zone cracked
hp
Steel sheeting, section Ap
I cc
bxc3 12 n
Tension zone uncracked
Steel sheeting centroid axis
Steel sheeting, section Ap
xc 2 ) 2 A (d x )2 I p p c p n
bxc (
Ea Ea n E 'cm 1 ( E Ecm ) cm 2 3
I cu
bhc3 12 n
hc 2 3 ) 2 bm .h p bm .h p (h x h p )2 t u n 12 n n 2
bhc ( xu
Ap (d p xu )2 I p
Summary • Composite slabs are widely used in steel framed buildings • Steel deck acts as - shuttering during construction - steel reinforcement for the concrete slab.
• Design of composite slabs requires consideration of two conditions 1. steel deck as a relatively thin bare steel section supporting wet concrete and construction operatives 2. as a composite structural element in service
Summary
â&#x20AC;˘ Performance of a composite slab is dependent on the effectiveness of the shear connection between the concrete and steel sheeting. â&#x20AC;˘ Longitudinal shear resistance may be assessed by: 1. A semi-empirical design method (m-k) 2. Partial interaction theory
Structural Steelwork Eurocodes 113
Shear Connectors and Structural Analysis
Scope of the lecture 114
Shear
connectors
– Generalities about shear connectors – Design resistance of usual shear connectors Structural
frame analysis
– Global analysis for Ultimate Limit States – Global analysis for Service Limit States
Shear connectors 115
Profiled Steel Sheeting
Shear connectors
Steel beam Transverse rebars
Shear force T in the connectors 116
T V: I : S:
V S ď&#x20AC;˝ I
vertical shear force in the beam second moment of area of the section first moment of area of either the concrete slab or the steel section about the elastic neutral axis.
Formula applicable in the elastic domain
Forces applied to connectors 117
ď ľ No
shear connection Beam section
Elastic range
Full plastic state Strains
Bending stresses
Shear stresses
Forces applied to connectors 118
ď ľ Full
connection Beam section
Elastic range Full plastic state Strains
Bending stresses
Shear stresses
Shear connectors 119
ď ľ Basic
forms of connectors
Stud connector
T connector
Angle connector
Hoop block connector
Shear connectors 120
P P (shear)
P
P Rk
Rk
slip su
Ductile connector
s
s
Non ductile connector
Criteria available in Eurocode 4
Shear connectors 121
ď ľ Deformation Slip
of flexible connectors
Crushed concrete
Shear connectors 122
ď ľ Rigid
and flexible connectors q
Connector force
Connector force
q
Distance along the beam
Distance along the beam
q = 0,7 times the plastic failure load q = 0,98 times the plastic failure load
Design resistance of studs 123
PRd min( PRd(1) , PRd( 2) ) PRd(1) 0,8. f u . ( . d ² / 4) / v
d
PRd( 2) 0,29 . . d ² . f ck . Ecm / v where : fu is the ultimate strength of the stud fck is the characteristic strength of the concrete is the corrective factor depending on h/d v is the partial safety factor
h
Design resistance of studs 124
For
headed studs with profiled steel sheeting the design shear resistance is multiplied by a reduction factor.
This
reduction factor depends on :
- the geometry of the slab - the relative position (II or ) of the steel beam and the sheeting ribs - the number of stud connectors in one rib
Design resistance of welded angle 125
The resistance of a welded angle connector is :
PRd
ď&#x20AC;˝
10 . l . h3 / 4 . f ck2 / 3
ď §v
minimum section required to avoid uplift
h
l
Global structural analysis 126
Analysis for ultimate – Rigid-plastic analysis – Elastic analysis
limit states
Analysis for serviceability – Global analysis – Calculation of deflections – Concrete cracking – Vibrations
limit states
Analysis for ultimate limit states 127
First-order
analysis allowed if:
VSd Vcr where:
VSd Vcr
0,1
is the design value of the total vertical load is the elastic critical load
non-sway frames
Analysis for ultimate limit states 128
Second-order
analysis required if:
VSd 0,1 Vcr Full set of application rules only available in Eurocode 4 for non-sway frames
main scope : non-sway frames
Analysis for ultimate limit states 129
Actual frame behaviour influenced by: – Cracking in hogging regions of beams – Local plasticity and subsequent redistributions – Slip between steel members and slabs – Possible uplift of the slab –… Elasto-plastic analysis required But not relevant for practical applications
Analysis for ultimate limit states 130
Two
analysis methods in Eurocode 4:
– Rigid-plastic analysis Plastic hinge theory – Elastic analysis Cracked analysis Uncracked analysis
Analysis for ultimate limit states 131
Rigid-plastic
analysis
– Ultimate frame resistance reached through the development of a full plastic mechanism – Plastic hinges form in critical sections (members or joints) where plastic rotation capacity is required – Conditions for application to be fulfilled (see two next slides)
Analysis for ultimate limit states 132
Restricted
field of application of the rigidplastic analysis : – The frame is non-sway – The frame, if unbraced, is of two storeys of less – All the members and joints of the frame are steel or composite – The cross-sections of steel members satisfy the principles of clauses 5.1.6.4 and 5.2.3 of EN 1993-11:20xx – The steel material satisfies clause 3.2.3 of EN 19931-1:20xx (continued)
Analysis for ultimate limit states 133
– At each plastic hinge location: The cross-section of the structural steel member or component should be symmetrical about a plane parallel to the plane of the web or webs The proportions and restraints of steel components should be such that lateral-torsional buckling does not occur Lateral restraint to the compression flange should be provided at all hinge locations at which plastic rotation may occur under any load case The rotation capacity should be sufficient to enable the required hinge rotation to develop Where rotation requirements are not calculated, all members containing plastic hinges should have effective cross-sections at plastic hinge locations that are in Class 1.
Analysis for ultimate limit states 134
The
rotation capacity in member crosssections is assumed to be sufficient for plastic redistribution if : – The grade of structural steel does not exceed S355 – All effective cross-sections at plastic hinge locations are in Class 1; and all other effective cross-sections are in Class 1 or Class 2
(continued)
Analysis for ultimate limit states 135
â&#x20AC;&#x201C; Each beam-to-column joint has been shown to have sufficient rotation capacity, or to have a design moment resistance at least 1,2 times the design plastic moment resistance of the connected beam â&#x20AC;&#x201C; Adjacent spans do not differ in length by more than 50% of the shorter span and end spans do not exceed 115% of the length of the adjacent span L 2 - L 1 < 0,50 L 1
L 1< L 2
L2
L 1 < 1,15 L 2
L1
L2
(continued)
Analysis for ultimate limit states 136
â&#x20AC;&#x201C; In any span in which more than half of the design load for span is concentrated within a length of onefifth of the span, then at any hinge location where the concrete slab is in compression, not more than 15% of overall depth of the member should be in compression (to avoid a premature collapse due to concrete crushing). This condition does not apply where it can be shown that the hinge will be the last to form â&#x20AC;&#x201C; The steel compression flange at a plastic hinge location is laterally restrained
Analysis for ultimate limit states 137
Elastic analysis – Actual composite cross-section replaced by an equivalent steel section – Use of an « equivalent modular ratio n » (see Lecture 3):
Ea n E ' c steel E is the elastic modulus for a
E’c is an elastic « effective »modulus for concrete
(continued)
Analysis for ultimate limit states 138
Rough
values of E’c for evaluation of n
– E’c = Ecm for short-term effects – E’c = Ecm / 3 for long-term effects Strength class of concrete
Ecm (kN/mm2)
20/25 25/30 30/37 35/45 40/50 45/55 50/60
29
30,5
32
33,5
35
36
37
(continued)
Analysis for ultimate limit states 139
ď ľ More
precise values of n
Options
Short-term effects
Long-term effects
Comments
(a)
Account of secant modulus Ecm
Various, depending on concrete grade
Account of concrete grade and age
(b) 6
18
(c)* 15
15
*restricted to beams with Class 1 or 2 sections
Account of concrete age, but not grade No account of concrete grade and age
Analysis for ultimate limit states 140
Elastic analysis – Loss of rigidity due to cracking of the concrete in hogging bending regions – Possible yielding and local buckling phenomena in steel members on supports two approaches in Eurocode 4: cracked elastic analysis uncracked elastic analysis
Analysis for ultimate limit states 141
ď ľ Uncracked
elastic analysis
â&#x20AC;&#x201C; Constant beam flexural stiffness (EaI1) based on an effective width b+eff of the slab evaluated at mid-span
Pd
Pd
L2
L1
Ea I 1 a) "uncracked" method
0,15 L 1 L1
EaI 1
x
0,15 L 2 L2
Ea I 2
a) "cracked" method
Ea I 1
Analysis for ultimate limit states 142
Cracked
elastic analysis
– Constant beam flexural stiffness (EaI1) in sagging moment regions – Constant beam flexural stiffness (EaI2) based on an effective width b-eff of the slab defined on support in hogging moment regions Pd
Pd
L2
L1
Ea I 1
0,15 L 1 L1
EaI 1
x
Ea I 2
0,15 L 2 L2
Ea I 1
Analysis for ultimate limit states 143
Redistribution
of internal forces allowed by Eurocode 4 further to an elastic analysis – To take into account cracking of concrete, inelastic behaviour of materials and all types of buckling – Principles: Decrease of the bending moments M peak on beam supports As a result, increase of the « in-span » moments Respect of the equilibrium with applied loads (continued)
Analysis for ultimate limit states 144
Maximum
redistribution percentage p
M Rd M Rd M peak 1 p/100) Class of cross-section (hogging bending)
1
2
3
4
Uncracked elastic analysis
40%
30%
20%
10%
Cracked elastic analysis
25%
15%
10%
0%
Analysis for serviceability limit states 145
Global analysis – Elastic analysis taking account of concrete cracking, creep and shrinkage – The member flexural stiffness is the uncracked one – Redistribution achieved because of cracking in hogging moment regions (3 possible methods in EC4) – Easiest redistribution method: Reduction of the maximum hogging moments Class of section (hogging bending)
1
2
3
4
For cracking of concrete
15%
15%
10%
10%
Analysis for serviceability limit states 146
Calculation
of beam deflections
– Elastic analysis – The member flexural stiffness is the uncracked one – Redistribution achieved because of concrete cracking and steel yielding in hogging moment regions 0 1 (M AM B) M0
0 and M0 M-A and M-B
are the deflection and sagging moment at mid-span if the beam is assumed to be simply supported is the redistribution factor are the bending moments at beam ends from the uncracked analysis
Analysis for serviceability limit states 147
Concrete
cracking
– Due to restrained deformations (shrinkage, displacement of support) and direct service loads – Cracking is checked in situations where it affects the function and durability of the structure Appearance criteria can also be a factor. – Design requirements available (continued)
Analysis for serviceability limit states 148
– When no specific measures are adopted to limit cracking, a minimun percentage of longitudinal reinforcement within the beam effective width is required 0,4% of the concrete area for a propped structure 0,2% of the concrete area for a unpropped structure
Steel sheeting not considered – The reinforcement bars are lenghten on a quarter of the span (in hogging moment regions) – When the width of the cracks has to be limited, provisions for minimum longitudinal reinforcement given in Eurocode 4
Analysis for serviceability limit states 149
Vibrations
– Check that the eigen frequencies of the structure or of parts of it are different from those at the source of vibrations (machines, …) – Eigen frequency for a composite floor: Uncracked properties of the section Possible slip at shear interface neglected Secant elasticity modulus Ecm for concrete (shortterm) Simply supported beam
g 1 f with g 9810mm/sec2 2 (continued)
Analysis for serviceability limit states 150
– Formula valid for simply supported composite beams – Also applicable for slabs overlapping several parallel composite beams with:
s b s is the deflection of the slab (with regard to the beam)
b is the beam deflection –
Critical frequencies – Normal floor: – Gymnastic or dance floor:
3 Hz 5 Hz
Structural Steelwork Eurocodes Development of a Trans-National Approach 151
Simply Supported Composite Beams
Principal Design Checks 152
Composite
beam design to EC4 is according to Limit State Design principles The principal checks are: – Ultimate Limit State (ULS) • Moment resistance • Shear connection • Vertical shear – Serviceability Limit State (SLS) • Deflection • Concrete cracking
Section Classification of Composite Beams 153
Local
buckling is controlled by section classification Sections are classified according to the least favourable class of its steel elements in compression – Flange – Web Class
of section and slenderness limits are identical to those for bare steel sections - EC3
Section Classification of Composite Beams 154
Class 1 and 2 – capable of developing the full plastic bending moment M+ – can also rotate after the formation of a plastic hinge
Class 3 – full plastic moment resistance cannot be achieved – stresses in the extreme fibres of the steel section can reach yield
Class 4 – Local buckling occurs before yield is reached
Classification of Compression Flange 155
Beams
acting compositely with concrete slab
– Compression flange is restrained from buckling by the concrete slab – Flange may be defined as Class 1 Partially
encased beams
– Infill between flanges provides incomplete restraint – Slenderness limits apply – Table 5.4
Classification of steel flanges EC4 Table 5.4 156
Limits for rolled and welded sections: Class Limit Rolled (Welded)
1 2 3
c/t c/t c/t
10 15 21
c = flange outstand; t = flange thickness;
9 14 20 = (235/fy)
Classification of Web 157
If
the plastic neutral axis is in the slab or the upper flange – Web is in tension throughout – Web is designated as Class 1
If
plastic neutral axis is in the web
– Slenderness of the web should be checked – Table 5.2a of EC3 – Seldom applies for simply supported beams
Modifications to the classification of the web 158
Class
3 webs may be reclassified if compression flange is class 1 or 2 – A class 3 web encased in concrete can be taken as a class 2 – A class 3 web not encased in concrete can be taken as an equivalent class 2 web • Based on an effective height of the web in compression • Two parts of the same height 20t
Plastic Moment Resistance of Class 1 or 2 Sections 159
Bending resistance based on plastic analysis Simplified assumptions
– Full interaction – All fibres of the steel beam are yielded in tension or compression – Compression stresses in the concrete are uniform and equal to 0.85fck/c – Concrete in tension is negligible – Slab reinforcement in tension is yielded with a stress of fsk/s – Slab reinforcement and the decking in compression have negligible effect
Composite beams with composite decking 160
For
composite beams with composite floor slabs, additionally assumptions apply – Concrete in the ribs is ignored – This limits the depth of concrete in compression
General
analysis of composite beams can be applied to solid floor slabs by setting the depth of profiles, hp, to zero
Plastic Neutral Axis Located in the Slab Depth 161
(compression) b hc
+ eff
0,85 f
P.N.A.
ck / c N cf F
z
hp ha / 2 Npla a
ha ha / 2 f y / a (tension)
Plastic Neutral Axis Located in the Slab Depth 162
Plastic
axial resistance of the steel beam (in tension) Npla Npla = Aafy/a where Aa is the area of the steel beam
Axial
resistance of the Concrete slab Ncf: Ncf = hcb+eff(0.85fck/ c) where b+eff is the effective width of the slab
Plastic Neutral Axis Located in the Slab Depth 163
Consider
longitudinal equilibrium Plastic neutral axis is located in the thickness hc if
Ncf > Npla Depth
of the plastic neutral axis z:
z = Npla/ (b+eff 0.85fck/ c) < hc Moment
resistance:
M+plRd = Npla (0.5ha + hc + hp - 0.5z)
Plastic Neutral Axis Located in the Flange of the Steel Beam 164
(compression)
N
cf N
N tf bf
(tension)
pla2
pla1
Plastic Neutral Axis in the Flange of the Steel Beam 165
If
Ncf < Npla
– plastic neutral axis is located in below the level of the interface within the upper flange of a symmetric steel beam – z is greater than the total thickness of the slab (hc + hp)
Plastic Neutral Axis in the Flange of the Steel Beam 166
Also
Npla1 < bf tf fy/a or
Npla - Ncf< 2bf tf fy/a Two
equal and opposite forces, Npla1 added
to:
Ncf + Npla1 - Npla2 = 0 Ncf + 2Npla1 - (Npla2 + Npla1) = 0
Plastic Neutral Axis in the Flange of the Steel Beam 167
Npla = Npla1 + Npla2 Npla1 = 0.5(Npla - Ncf) Npla = Ncf + 2Npla1
Noting
Depth
of the flange in compression is
[z - (hc + hp)] Npla1 = b1 (z - hc - hp)fy/a, Npla = Ncf + 2b1 (z - hc - hp).fy/a
Plastic Neutral Axis in the Flange of the Steel Beam 168
ď ľ Taking
moments about the centre of gravity of the concrete:
M+pl..Rd = Npla(0.5ha + 0.5hc + hp) - 0.5(Npla - Ncf)(z + hp)
Plastic Neutral Axis in the Web of the Steel Beam 169
N
cf N
w
P.N.A. t w
pla1 N
pla2
(tension)
Plastic Neutral Axis in the Web of the Steel Beam 170
If, simultaneously:
Ncf > Npla and Npla - Ncf > 2bf tf fy/a Neutral axis is located within the beam web Tensile force Npla1 is balanced by an equal and opposite force This acts in equivalent position on opposite side of beam centre of gravity Remaining tensile force balances Ncf
Plastic Neutral Axis in the Web of the Steel Beam 171
This
force to balance Ncf
– acts over a depth of the web 2zw – at a stress of fy/a – at centre of gravity of steel beam – equilibrium gives zw = Ncf/(2tw fy/a)
Moment
of resistance: M+pl.Rd = Mapl.Rd + Ncf(0.5ha + 0.5hc + hp) - 0.5 Ncf zw
Moment Resistance for Class 3 Sections 172
Slender
webs Two methods - elastic or plastic Plastic moment resistance M+pl.Rd Replace class 3 web with an equivalent class 2 web M+pl.Rd is then calculated as above Elastic
moment resistance M+el.Rd
Full steel section
Elastic Moment Resistance (Class 3 Sections) 173
Based on elastic modulus of transformed section Effects of creep in concrete important
– Duration of load – Propped and unpropped construction – Special considerations for storage buildings
Composite cross-section is not symmetrical – Consider two section moduli – Extreme top and bottom of section
Modular Ratio for Non-Sway Structures not for Storage 174
Modular
ratio, n, is the ratio of modulus of elasticity for steel and concrete n = Ea / Ec An average value, nav may be used Corresponds to an effective modulus of elasticity for concrete, Ec = Ecm/2 Ecm is the secant modulus of elasticity for concrete for short-term loading
Modular Ratio for Non-Sway Structures not for Storage 175
Ea = 205N/mm2 for steel and Ecm from Table 3.2 of EC2 Average values of modular ratio, nav are: Using
fck at 28 days 25 30 35 (N/mm2) nav 13.8 13.1 12.5
Propped Composite Beams in Structures not for Storage 176
For
propped construction the beam functions exclusively as composite
Elastic
moduli of composite section
– Wc.ab.el bottom of the steel beam – Wc.at.el top of the steel beam – Wc.ct.el upper fibre of the concrete slab
Propped Composite Beams in Structures not for Storage 177
of resistance M+el.Rd is then the minimum of:
Moment
Wc.ab.el fy/a Wc.at.el fy/a Wc.ct.el (0.85fck)/c
Elastic moment resistance unpropped construction 178
Define: – ab (at) the total longitudinal stress in the lower (upper) fibre of the steel beam – ct the total longitudinal stress in the upper fibre of the concrete slab – r the ratio of the total longitudinal stress to the permissible stress – Ma.Sd (Mc.Sd) the design bending moment before and after composite action developed
Stresses in Unpropped Structures not for Storage 179
Consider stresses in top and bottom of steel beam, and top of concrete slab. M a.Sd M c.Sd Thus: ab Wa ,ai,el Wc ,ab,el ( with n av )
M a.Sd M c.Sd at Wa ,at,el Wc ,at,el ( with nav ) M c.Sd ct Wc ,ct ,el
Stresses in Unpropped Structures not for Storage 180
These stresses must all be less than the corresponding material design strengths
ab Thus: rab 1,0 fy / a at rat 1,0 fy / a rct
ct
0,85. f ck / c
1,0
Elastic Moment Resistance in Unpropped Structures not for Storage 181
The
critical ratio is the maximum of each of these rmax = max{rab; rat; rct} The elastic moment resistance is then: M+el.Rd = (M0,Sd + ML,Sd) / rmax
Local Buckling in Unpropped Structures not for Storage 182
Limit effective slenderness of top flange according to EC3 Use actual stress due to self weight and construction loads, multiplied by a.
Thus:
p at a / cr 0,673
Warehouses and buildings used principally for storage 183
Must
account for the effects of concrete creep Modular ratio depends on type of loading
nL = n0(1 + L t) n0
= Ea/Ecm modular ratio for short term loading Ea is the modulus of elasticity for structural steel
Warehouses and buildings used principally for storage 184
Ecm
is the secant modulus of elasticity of the concrete for short term loading (Table 3.1 of EC2) t is the creep coefficient (EC2) L is the creep multiplier – may be taken as 1.1 for permanent loads
Warehouses and buildings used principally for storage 185
ď ľ Common
to use the strength of concrete at 28 days ď ľ Values of the modular ratio, n: fck at 28 days (N/mm2) Short term modular ratio n0 Long term modular ratio nL
25 6.9
30 6.6
35 6.3
20.7 19.7 18.8
Propped Composite Beams in Storage Buildings 186
Account
for different modular ratios, and recalling: – ab (at) the total longitudinal stress in the lower (upper) fibre of the steel beam – ct the total longitudinal stress in the upper fibre of the concrete slab – r the ratio of the total longitudinal stress to the permissible stress – M0.Sd (ML.Sd) the design bending moment due to short term and long term loading respectively
Stresses in Propped Composite Beams in Storage Buildings 187
ab
M 0.Sd M L.Sd Wc ,ab,el ( with n 0 ) Wc ,ab,el ( with n L )
M 0.Sd M L.Sd at Wc ,at,el ( with n 0 ) Wc ,at,el ( with n L ) M 0.Sd M L.Sd ct Wc ,ct ,el ( with n 0 ) Wc ,ct ,el ( with n L )
Stresses in Propped Composite Beams in Storage Buildings 188
The corresponding stress ratios are: ab rab 1,0 fy / a at rat 1,0 fy / a rct
cts
0,85. f ck / c
1,0
Elastic Moment Resistance in Propped Storage Buildings 189
The
critical ratio is the maximum of each of these rmax = max{rab; rat; rct} The elastic moment resistance is then: M+el.Rd = (M0,Sd + ML,Sd) / rmax
Unpropped Composite Beams in Structures for Storage 190
As for general buildings, but two modular ratios must be used ab
M a.Sd M 0.Sd M L.Sd Wa ,ab,el Wc ,ab,el ( with n 0 ) Wc ,ab,el ( with n L )
M a.Sd M 0.Sd M L.Sd at Wa ,at,el Wc ,at,el ( with n 0 ) Wc ,at,el ( with n L ) M 0.Sd M L.Sd ct + Wc ,ct ,el (with n 0 ) Wc ,ct ,el (with n L )
Unpropped Composite Beams in Structures for Storage 191
Stress ratios
ab rab 1,0 fy / a at rat 1,0 fy / a rct
ct
0,85. f ck / c
1,0
Elastic Moment Resistance in Unpropped Storage Buildings 192
ď ľ The
critical ratio for steel is ra = max{rab; rat} ď ľ The elastic moment resistance is the lesser of: M+el.Rd = (Ma,Sd + M0,Sd + ML,Sd) / ra M+el.Rd = (M0,Sd + ML,Sd) / rct
Shear Resistance 193
For
webs not prone to shear buckling: Shear stresses assumed to be carried by web of steel beam alone EC3 rules apply
Vsd < VplRd VplRd = Aw(fy/3) / a Aw is the shear area of the bare steel beam
Shear resistance - conditions for no shear buckling 194
For
unstiffened webs, not encased d/tw < 69 For unstiffened, encased webs d/tw < 124 For stiffened webs, not encased
.Ea tw 2 cr k ( ) 2 12.(1 ) d 2
k4+5,34/(a/d)² if a/d 1 k5,34+4/(a/d)² if a/d >1
Connector design - beams class 1 or 2 195
'critical
lengths' of beam between adjacent critical cross-sections: point of maximum bending moment supports concentrated loads Q A
L/2
L/2
d
C
B
L
Connector design - beams class 1 or 2 196
The
total longitudinal shear force VIN is:
VlN = min(Aa fy / a; 0,85beff hc fck /c) For
ductile connectors all assumed to be at the same load, PRd, the design strength of a single connector. The number of connectors for the critical length is therefore:
Nf(AB) = Nf(BC) = VlN / PRd spaced
uniformly over each critical length
Partial Interaction 197
Partial
interaction allowed for
– ductile connectors – class 1 or 2 sections Stud
connectors defined as ductile if:
– length not less than 4x diameter – 12mm < diameter < 25mm – Degree of shear connection, (= N/Nf), is greater than prescribed limits
Partial Interaction - minimum shear connection 198
Solid slab, equal steel flanges Lc<25m: > 1-(355/fy)(0,75-0,03Lc)
Solid slab, unequal steel flanges Lc<20m: > 1-(355/fy)(0,30-0,015Lc)
Composite slabs Lc<25m: > 1-(355/fy)(1-0,04Lc)
In all cases > 0,4
For
critical lengths longer than indicated
> 1
Partial Interaction - moment resistance 199
Reduced longitudinal shear force transferred between steel and concrete
V1red = NPRd < V1N Hence
moment resistance reduced
M+Rdred < M+plRd Moment resistance calculated using same principles as full interaction Stress blocks reduced to V1red in both steel and concrete
Graph of M+Rdred vs number of connectors 200
Neutral axis of the section (red) M pl.Rd
in the steel web
in the steel flange C
M pl.Rd
B
DUCTILE CONNECTORS M apl.Rd
A
(
N ) min Nf
1.0
N Nf
Connector design for class 3 or 4 beams 201
Based
on elastic behaviour Longitudinal shear stress V is given by:
V = T S1 / l Connectors
spaced to reflect shear
distribution – more closely spaced near supports
Transverse reinforcement 202
Total
area of transverse reinforcement per unit length crossing potential shear failure surface = Ae Total length of potential failure surface = Ls Design shear per unit length, VSc1, must not exceed shear resistance VRd of failure surface.
Transverse reinforcement a
a
203
At
At
A bh a a
a
Plane
b
b
a
Ab
b-b c-c
2 ( A b+ A bh )
d-d
Ab
Ae
A +At b 2 Ab
a-a
c
c
2 A bh
A bh
a
At
a d
d
Ab
Transverse reinforcement 204
VSc1
< VRd VRd = min(VRd(1) , VRd(2)) VRd(1) = 2,5Rd Ls + Ae fsk / s VRd(2) = 0,2 Ls fck / Rd Rd is the design shear strength for concrete fck
20
25
30
35
40
45
50
Rd
0,26
0,30
0,34
0,38
0,42
0,46
0,50
Transverse reinforcement composite slabs 205
Profiled
sheeting can be considered as equivalent reinforcement VRd(1) = 2,5Rd Ls + Ae fsk / s + Ap fp / ap where Ap - area of sheeting crossing failure surface fp - nominal elastic limit of sheet ap - appropriate partial safety factor (taken as 1,1)
Serviceability Limit State Design 206
Concerns:
– deflections – cracking of concrete – vibrations For
conventional buildings, rigorous analysis often unnecessary No serviceability limits imposed for stresses
Deflections 207
Calculated
deflections based on transformed section Deflection limits as for bare steel (EC3) Simplified approaches using limiting span:depth ratios For simply supported beams the limits are: – 15 to 18 for main beams – 18 to 20 for secondary beams (joists)
Concrete cracking 208
EC2
procedures may be adopted A simplified alternative is to – provide minimum reinforcement – limit bar spacing
Concrete cracking - simplified rules for minimum reinforcement 209
Minimum area of reinforcement, As:
As = ks kc k fct,eff Act / s fct,eff - mean tensile strength of concrete (3N/mm2) k 0,8; ks 0,9 kc = 1 / {1 + hc / (2 zo)} + 0,3 1,0 hc is the thickness of the concrete flange zo - distance between centroids of concrete flange and composite section Act - area of effective width of concrete s = fsk but lower values may apply (Table 5).
Concrete cracking - Values of s 210
steel stress s N/mm2 160 200 240 280 320 360 400 450
max bar diameter (mm) for design crack width wk = 0,4mm wk = 0,3mm wk = 0,2mm 40 32 25 32 25 16 20 16 12 16 12 8 12 10 6 10 8 5 8 6 4 6 5 -
Concluding summary 211
Sections are classified as for bare steel sections, but webs may be reclassified. The moment resistance of class 1 and 2 sections is calculated using plastic analysis The moment resistance of class 3 sections is calculated using elastic analysis Vertical shear strength is as the bare steel section
Concluding summary 212
Longitudinal shear connection is based on the force transmitted between the steel section and concrete slab. If connectors are insufficient beam may be designed as partially composite Deflection limits are as stated in EC3 Concrete cracking can be controlled by appropriate slab reinforcement
Structural Steelwork Eurocodes 213
Continuous Beams
Introduction 214
Continuous beams may be more economical than simply supported beams. However, special phenomena may occur which must be taken into account in design, such as: – local buckling of compressed plate elements – lateral-torsional buckling – cracking of concrete due to tensile stresses
These all occur in the hogging moment regions. In the sagging moment regions, design checks are similar to those of simply supported beams. See previous lecture.
Introduction 215
ď ľ
In the construction phase, hogging moment regions may extend to a larger part of a span than in normal conditions.
Negative moment
(a) Both spans loaded
Negative moment
(b) One span loaded
Part 1 216
Rigid-plastic Design
Rigid-plastic design 217
Rigid-plastic design consists of the following steps: – Analysis: rigid-plastic (see Lecture 4) – Design of cross-section – class 1 sections with appropriate moment resistance
We discuss the following problems: – How to determine the required plastic moment resistance of the cross-sections for a given load – How to classify cross-sections – How to determine the actual plastic moment resistance of a crosssection
Required plastic moment resistance 218
Assume we know from experience the ratio, , of negative to positive moments of resistance of the cross-section Example: end span of a continuous beam Wf L
1 1
L
Mpl = M'pl Mpl
M pl
w f 2 L2 2
Cross-section classification 219
Classification of cross-sections according to EC3 and EC4: – Class 1: Plastic moment resistance and large plastic rotations can develop – Class 2: Plastic moment resistance can develop, but rotation capacity is limited due to local buckling – Class 3: Moment resistance is limited to the elastic moment resistance because of local buckling – Class 4: Moment resistance is limited to a value below the elastic moment resistance because of premature local buckling
Local buckling may occur in the compression plates only Classification is based on width-to thickness ratios of compression plate elements of the steel section
Cross-section classification 220
Classification boundaries for flanges in uniform compression
Class Type 1 2 3
Rolled Welded Rolled Welded rolled Rolled Welded c
t
Web uncased (EC3) 9 9 10 10 c/t 14 14 welded
c
Web encased (EC4) 10 9 15 14 21 20 235 N/mm 2 t fy
Cross-section classification 221
Classification boundaries for webs in pure bending and uniform compression (EC3) Class 1 2 3
t
Pure bending 72 83 124
rolled welded d / t rolled d/t welded
Uniform compression 33 38 42
t
235 N/mm 2 fy
Cross-section classification 222
Class of the section is defined as class of the element with the less favourable behaviour (e.g.: class 1 web and class 2 flange = class 2 section)
Exception: if compression flange is at least class 2 and web is class 3, then the section can be considered class 2: – with the same cross-section, if the web is encased – with an effective web, if the web is not encased
b
eff
hc hp 20t w
d tw
20t w
Plastic cross-section resistance 223
Basic assumptions: – full connection between steel and concrete – all fibres yielded – steel & reinforcement: full yield strength – all concrete is in tension; resistance of concrete in tension is zero
Two main cases for which formulae are developed – case 1: plastic neutral axis is in the flange of the steel section – case 2: plastic neutral axis is in the web of the steel section
Plastic cross-section resistance 224
Case 1: plastic neutral axis is in the flange of the steel section tension
hs
P.N.A. tf
f
bf compression
M pl . Rd Fa (0,5ha hs ) ( Fa Fs )(0,5 z f hs )
Fs As f sk / s Fa Fs 2b f z f f y / a
Plastic cross-section resistance 225
Case 2: plastic neutral axis is in the web of the steel section tension f / s sk
b eff
F
hc
s
hp Fa
P.N.A. ha
zw tw
ha /2
Fa f y / a f y / a
M pl .Rd M apl .Rd Fc (0,5ha 0,5hc h p ) 0,5Fc z w
a Fs zw 2t w f y
Fs As f sk / s
Part 2 226
Elastic Design
Elastic design 227
Elastic design consists of the following steps: – Analysis: elastic – Design of cross-section – any class of cross-sections with appropriate moment resistance
We discuss the following problems: – Overview of problems related to analysis – Classification of cross-sections – Calculation of the elastic moment resistance of a cross-section
Elastic analysis – Overview 228
Effective width – although in reality the effective width depends on whether hogging or sagging moment regions are considered, a good approximation for global analysis is to consider a constant effective width, which is » that belonging to the sagging moment for the general case, » that belonging to the hogging moment for cantilevers
Analysis methods – Cracked analysis – Uncracked analysis
Elastic analysis – Overview 229
Redistribution of moments obtained from elastic analysis – Unlike for bare steel beams, moment redistribution is allowed for any class depending on the method of global analysis – The amount which is allowed to be redistributed depends on: » the class of the cross-section for hogging bending » the method of global analysis (cracked or uncracked)
Class “Uncracked” analysis “Cracked” analysis
1 40% 25%
2 30% 15%
3 20% 10%
4 10% 0%
Classification of cross-sections 230
Already discussed in the context of plastic design.
Elastic cross-section resistance 231
Basic assumptions: – similar to those of plastic resistance, except that elastic distribution of stresses is considered
Two cases: – Propped construction: all loads are carried by the composite cross-section – Unpropped construction: » loads applied prior to concrete hardening, are carried by the steel section alone » loads applied after concrete hardening, are carried by the composite section
Elastic cross-section resistance 232
Case 1: Propped construction M el .Rd
Wc.ab.el f y Wc.ss.el f y min ; a s
Stress in steel section limited to yield strength
Stress in reinforcement limited to yield strength
Elastic cross-section resistance 233
Case 2: Unpropped construction
Stress in extreme fibres of steel
ab
Stress in reinforcement
M a.Sd M M M c.Sd at a.Sd c.Sd Wa.ab.el Wc.ab.el Wa.at .el Wc.at .el
ss
M c.Sd Wc.ss.el
Utilisation ratios
a at a ab ra max ; fy fy
Resistance
M el .Rd
1,0
rs
s ss 1,0 f sk
M a.Sd M c.Sd M c.Sd min ; M a.Sd ra rs
Part 3 234
Problems Common to Both Elastic and Plastic Design Shear resistance Lateral-torsional buckling Shear connection design Loss of serviceability due to cracking
Resistance against combined bending and shear 235
Interaction diagram
Low bending – shear capacity not reduced
High bending and shear – interaction formula M v.Rd M f .Rd ( M Rd
V Sd V pl.Rd
C
2 2V M f .Rd ) 1 Sd 1 V pl .Rd
B
0,5 V pl.Rd
A
_ M
f.Rd
_ M
Rd
Moment capacity of flanges only
_ M
V.Rd
Low shear – moment capacity not reduced
Lateral-torsional buckling 236
The theory of lateral-torsional buckling of continuous beams over supports is rather complex.
In reality, lateral-torsional buckling is affected by: – beam distortion / lateral deflection of compressed flange – torsional rigidity of section
In design, two types of simplified approach may be followed: – simplified calculation of lateral-torsional buckling resistance according to analogy to steel beams (EC3 approach) – application of certain detailing rules that prevent lateral-torsional buckling
Lateral-torsional buckling 237
EC3 approach
M b.Rd
LT M Rd
EC3 LT buckling curves
LT
M pl M cr
In this approach, the elastic critical moment is determined using the so-called “inverted U-frame model”. The use of this model is subject to certain conditions. This model is not discussed here in detail
Lateral-torsional buckling 238
Prevention of lateral-torsional buckling by detailing rules
These rules refer to:
the regularity of adjacent span lengths the loading of the spans and the share of permanent loads the shear connection between top flange and concrete slab the neighbouring member supporting the slab the lateral restraints and web stiffeners of the steel member at its supports the cross-sectional dimensions of the steel member the depth of the steel member (depending on section shape, steel grade and presence of encasement)
Shear connection design 239
Basic rules – Connectors should be ductile – Plastic design of shear connection is possible even if global analysis is elastic, provided that the end cross-sections of the critical length to be designed are at least Class 2 – In hogging moment regions, use of full shear connection is recommended – In sagging moment regions, partial shear connection may be applied
Shear connection design – Example 240
Problem: Design of shear connection for a continuous beam with Class 1 crosssections (rigid-plastic analysis), assuming ductile connectors M u'
L Q A
B
d
M
C
(red) u
Ultimate load M u( red ) L M u d Q d (L d )
…where Mu(red) depends on the degree of shear connection
Shear connection design – Example 241
Equilibrium of slab
For section AB:
bending diagram A M
M
B
+(red) pl.Rd
+
A
Vl( AB) N ( AB) PRd Fu( red ) pl.Rd
C
For section BC: Vl( BC ) N ( BC ) PRd Fu( red ) Fs
B -F (red) (AB) +- V L
F (red)
Fu( red ) N ( BC ) PRd Fs
B
C
-F (red) F (red)
Fs
(BC) +- V L
Total number of connectors:
-F s
M u(red )
N N ( AB) N ( BC ) 2 N ( BC ) FS / PRd
Shear connection design – Example 242
Number of connectors necessary for a full connection over BC: Ultimate load in the function of total number of connectors
N (f BC )
1 PRd
Q Qu 1,0
Aa f y 0,85beff hc f ck As f sk min ; c s a
Plastic hinge theory
A'
B'
C'
0
( N/N f ) B'
1,0
N/N f
Serviceability – Cracking of concrete 243
This limit state is particular for continuous beams. (For other serviceability problems, see lecture on simply supported beams)
In simply supported beams, concrete may crack due to shrinkage
In continuous beams, concrete cracking is mainly due to tensile stresses in the hogging moment regions
This cracking is prevented by limiting bar spacing or bar diameters in the reinforcement
Serviceability – Cracking of concrete 244
Limiting bar spacing (for high bond bars only) to avoid cracking over supports stress in reinforcement s, N/mm2 160 200 240 280 320 360
maximum bar spacing for wk = 0,4 mm 300 300 250 250 150 100
this stress is calculated considering tension stiffening
maximum bar spacing for wk = 0,3 mm 300 250 200 150 100 50
maximum bar spacing for wk = 0,2 mm 200 150 100 50 – –
unless using a more precise method: s s 0 s
…with... s
0.4 f ctm st s
Conclusions 245
Continuous beams offer advantages over simply supported beams, but special phenomena need particular attention during design in the hogging moment regions In the case of both elastic and plastic design, cross-section classification and resistance calculation are key issues Lateral-torsional buckling at the hogging moment regions must be prevented by appropriate detailing or by direct check In shear connection design, hogging moment regions require full shear connection In the hogging moment regions, the serviceability limit state of cracking of concrete may be relevant
Structural Steelwork Eurocodes
Composite Columns
General comments on composite columns
Can be complex in fabrication and/or construction,
but …
Can be very strong - range of capacities for the same external dimensions. It may be possible to keep columns externally similar over all storeys of a building.
Most types have high inherent fire resistance without additional protection.
Concrete-encased sections bc Completely Encased Steel Section
b cy
cy Concrete usually provides all necessary fire resistance
cz
h
y
tw
t f
cz z
hc
Concrete-encased sections
Partially Encased Steel Section
b = bc
Concrete is poured in 2 stages with section horizontal. Needs additional reinforcement for fire resistance. y May need additional fire protection material.
h = hc
tw
t f
May need studs or rebars welded to section for force transfer.
z
Concrete-encased sections
Fabricated Steel Section
b = bc b
Concrete may be pumped into voids during construction.
h = hc
y tw
t f
z
Concrete-filled hollow sections
Concrete-Filled Rectangular Hollow Section
b t
Concrete may be pumped into hollow section during construction. Confined concrete has higher strength than in normal use.
h
y t
Needs additional reinforcement for fire resistance. May need additional fire protection material.
z
Concrete-filled hollow sections
Concrete-Filled Circular Hollow Section
d
Concrete may be pumped into hollow section during construction. Confined concrete under hoop tension has much higher strength than in normal use.
t
y
Needs additional reinforcement for fire resistance.
May need additional fire protection material.
z
Concrete-filled hollow sections
Concrete-Filled Circular Hollow Section encasing an open section
d
The internal steel section can enhance strength to a very high level.
t
y
z
General and Simplified design methods General Method • Second-order effects and imperfections taken into account in calculation, • Can be used for asymmetric sections,
• Needs suitable numerical software.
Simplified Method • Full interaction between the steel and concrete sections until failure, • Geometric imperfections and residual stresses taken into account in calculation, usually using European buckling curves,
• Plane sections remain plane.
Avoiding local buckling - fully encased sections
Concrete cover to section (cy) :
cy
b
cy
•
must be reinforced laterally,
•
> 40mm
•
> b/6
Avoiding local buckling - partially encased/concrete filled sections 235 / fy.k
where fy.k is characteristic strength of section
b
d
t
b
d t tf
d / t 90 2
d / t 52
b / t f 44
Force transfer in a composite beam-column connection Maximum Shear between Steel Section and Concrete: Partially encased sections 0,2 N/mm2 flange 0 N/mm2 web
Transfer length p < 2,5d
Completely encased sections
0,3 N/mm2
d
Concrete-filled hollow sections 0,4 N/mm2
Fin plates welded to the column section
Use of studs to enhance force transfer in composite columns mPRd/2
PRd
If insufficient shear capacity in transfer length, use studs to carry the remaining part of the force transferred to the concrete:
ď &#x2014;300mm Additional shear on inside of each flange = mPRd/2. Assume m=0,5 initially, but really depends on confinement.
Use of studs to enhance force transfer in composite columns mPRd/2
PRd
PRd
If insufficient shear capacity in transfer length, use studs to carry the remaining part of the force transferred to the concrete:
ď &#x2014;400mm Additional shear on inside of each flange = mPRd/2. Assume m=0,5 initially, but really depends on confinement.
Use of studs to enhance force transfer in composite columns mPRd/2
PRd
PRd
PRd
If insufficient shear capacity in transfer length, use studs to carry the remaining part of the force transferred to the concrete:
ď &#x2014;600mm Additional shear on inside of each flange = mPRd/2. Assume m=0,5 initially, but really depends on confinement.
Simplified design method Limitations General Limitations Columns prismatic and symmetric about both axes over the whole height, 5,0 > (depth/width) > 0,2,
Steel section to carry between 20% - 90% of the design resistance of the composite section, Relative slenderness 2,0;
of the composite column must be less than
Simplified design method Concrete-encased sections Longitudinal reinforcement area 0,3% of concrete cross-section area.
bc cy
cy
cz
Concrete cover : y-direction: 40 mm cy 0,4 bc z-direction: 40 mm cz 0,3 hc
y
hc
Only include area of longitudinal reinforcement in calculating crosssectional resistance up to 6% of the area of the concrete.
cz
z
Axial Compression - Cross-section Resistance Cross-section resistance to axial compression is the sum of the plastic compression resistances of each of its elements:
Concrete-encased sections Npl.Rd A a
fy Ma
fck fsk A c .0,85 A s c s
Section Concrete Reinforcement
Axial Compression - Cross-section Resistance Cross-section resistance to axial compression is the sum of the plastic compression resistances of each of its elements:
Concrete-filled hollow sections
Npl.Rd A a
fy Ma
fck fsk Ac As c s
Section Concrete
Confinement causes increased concrete resistance from 0,85fck to fck.
Reinforcement
Axial Compression - Cross-section Resistance Concrete-filled circular hollow sections d
More concrete compressive resistance is caused by hoop stress in the steel section. Only happens when most of the lateral expansion of concrete is prevented.
t Used in design if: 0,5
•
Relative slenderness
•
Maximum bending moment
Mmax.Sd 0,1NSdd
Axial Compression - Cross-section Resistance Concrete-filled circular hollow sections Plastic compression resistance is:
Npl.Rd A aa
fy Ma
fck Ac c
fsk t fy 1 c d f A s ck s
If equivalent eccentricity e=Mmax.Sd /NSd then Section Reinforcement for 0<ed/10 Concrete e e a a0 (1 a0 )10 c c 0 (1 10 ) d d 2 c 0 4,9 18,5 17 0 a0 0,25(3 2) 1,0 Eccentricity effect
Slenderness effect
For Mmax.Sd /NSd > d/10 use a=1,0 and c=0
Elastic critical load of a composite column Elastic critical load
2 (EI)e Ncr L2fl
For short-term loading Effective stiffness
(EI)e EaIa 0,8
Ecm Ic EsIs c
Ecm secant modulus of concrete
c
Partial safety factor for concrete stiffness (=1,35)
0,8 Reduction factor for cracking
Lfl is buckling length of column (may be taken as system length for rigid frame).
Elastic critical load of a composite column Elastic critical load
2 (EI)e Ncr L2fl Ec Ecm
For long-term loading Effective stiffness
(EI)e EaIa 0,8EcIc EsIs
1 NG.Sd 1 t NSd
NG.Sd is permanent part of the axial design load NSd
t
is EC2 creep coefficient
Lfl is buckling length of column (may be taken as system length for rigid frame).
Relative slenderness of a composite column
Npl.Rk
(Npl.Rk is Npl.Rd calculated using a = c = y = 1,0 )
Ncr
Long-term Concrete Modulus only modified to if
Ec Ecm
0,8
for concrete-encased sections 0,8 /(1 ) for concrete-filled hollow sections
where
A a fy Ma Npl.Rd
1 N 1 G.Sd t NSd
is the relative contribution of the steel section to overall axial plastic resistance.
These limiting values apply in the case of braced non-sway frames. 0,5 and 0,5 /(1 ) for sway frames.
and relative eccentricity e/d < 2,0 in the plane of bending considered.
Buckling resistance of a composite column - Strength reduction factor Buckling reduced from critical by factor
1 2 1/ 2
[ ] 2
1 in which
Buckling curves for composite columns:
0,5[1 ( 0,2) ]
Nb.Rd / Npl.Rd
Impf. Column Type
(a) 0,21 L/300 Concrete-filled sections, reinf < 3%, no steel section.
2
Plastic resistance 1,0
Perfect critical loads
(b) 0,34 L/210 Encased H-sections in major axis buckling, Concrete-filled sections, 3%<reinf<6%, or withsteel section. (c) 0,49 L/170 Encased H-sections in minor axis buckling.
0
1,0 Relative Slenderness
Resistance of cross section under axial force and uniaxial bending Conc.
N
Sec. Rft. Npl.Rd
A Npl.Rd 0,85fck/c
Point A: Axial compression resistance
M 0
fy/Ma fsk/s
Resistance of cross section under axial force and uniaxial bending Conc.
N 2hn
A
Sec. Rft. + +
Npl.Rd 0,85fck/c
+
Mpl.Rd
fy/Ma fsk/s
Point B: Uniaxial bending resistance
B 0
Mpl.Rd
M
Resistance of cross section under axial force and uniaxial bending Conc.
Sec. Rft.
N 2hn
A
+
Npl.Rd 0,85fck/c
Npm.Rd
Mpl.Rd
+
Mpl.Rd
fy/Ma fsk/s
Point C: Uniaxial bending resistance with nonzero axial compression
C
B 0
Npm.Rd
M
Resistance of cross section under axial force and uniaxial bending Simplest resistance locus
N
Conc.
A
+
Npl.Rd
+
0,85fck/c
Npm.Rd
fy/Ma
Npm.Rd/2 fsk/s
Point D: Maximum bending resistance
C
0,5Npm.Rd
D B
0
Sec. Rft. Mmax.Rd
Mpl.Rd Mmax.Rd
M
Resistance of cross section under axial force and uniaxial bending More complex resistance locus
N
Conc.
A
N
+ +
Npl.Rd
E 0,85fck/c
Npm.Rd
+
fy/Ma fsk/s
Point E: 50% of uniaxial bending resistance
C
0,5Npm.Rd
D B
0
Sec. Rft. Mpl.Rd/2
Mpl.Rd Mmax.Rd
M
Resistance of cross section under axial force and uniaxial bending Real resistance locus N A Npl.Rd
There is little advantage in using the real resistance locus in most cases.
E
AECDB may be more useful than ACDB if axial force is high.
C
D B 0
Mpl.Rd
M
Second-order amplification of bending moments Maximum bending moment is amplified by the second-order effect of axial force and deflection.
NSd First-order bending moments
M
It is only necessary to amplify moments if: NSd / Ncr 0,1 and
Second-order bending moments
0,2(2 r )
= 0,66+0,44r if subject to end moments, = 1,0 if lateral loads.
Amplification factor k
1 NSd / Ncr
1,0
NSd
rM
Resistance of column under axial force and uniaxial bending N/ Npl.Rd
Resistance locus of the cross-section 1,0 ď Łd=NSd/Npl,Rd M may be determined using amplification for second-order effects if the slenderness and axial force fulfil the previous criteria.
Limiting value MSd/Mpl,Rd ď &#x2014; 0,9md md=MRd/Mpl,Rd 0
1,0
M/ Mpl.Rd
The effect of shear on bending resistance • It can be assumed that the transverse shear Vsd is carried by the steel section only. • The effect of shear only needs to be taken into account if the shear force is more than 50% of the shear resistance Vpl.a.Rd of the steel section. • Thickness of the shear area is reduced over the sheared zone (usually the web of the steel section). The reduction factor is: 2 2V a.Sd w 1 1 Va.Rd
• The reduced shear area for an H-section bending about the major axis is: w t w .h
Resistance of column under axial force and biaxial bending N • (y-y) “Buckling” Axis
My
• (z-z) “Stronger” axis My N y
Mz N
Mz N
z
z
y
Resistance of column under axial force and biaxial bending N/Npl.Rd
“Buckling” axis: amplify My for imperfection.
1,0 NSd Npl,Rd
N/Npl.Rd 1,0 NSd Npl,Rd
0,9mdy 0
0,9mdz mdz Mdz.Sd/Mpl.z.Rd
“Strong” axis: no imperfection amplification.
0,9mdz 1,0 mdy
My/Mpl.y.Rd
My.Sd/Mpl.y.Rd
mdz 1,0
0
Mz /Mpl.z.Rd
My
Mz
Resistance of column under axial force and biaxial bending At a constant axial compression force NSd the
Design moments
0,9mdy mdy
My.Sd/Mpl.y.Rd
My.Sd 0,9mdyMpl.y.Rd 0,9mdz mdz
Mdz.Sd/Mpl.z.Rd
My.Sd m dyMpl.y.Rd
Mz.Sd 1,0 m dzMpl.z.Rd
Mz.Sd 0,9mdzMpl.z.Rd
Structural Steelwork Eurocodes 283
Composite Joints
Introduction 284
Types of joints Construction of joints used to be based on the already well-known conventional steeljoints:
Introduction 285
Types of joints Conventional
joints
Simple joints M
pinned
Introduction 286
Types of joints Conventional
joints
Continuous joints M
rigid, full strength
Introduction 287
Types of joints Advanced
joints
Semi continuous joints M Resistance
stiffness
rotation capacity
rigid or semi-rigid, full or partial strength, specific rotation capacity
Introduction 288
Types of joints conventional and advanced CONVENTIONAL - steel joints
gap
welded
gap
angle cleats
gap
flush endplate
welded
angle cleats
flush endplate
Introduction 289
Types of joints conventional and advanced ADVANCED - composite joints
b a l s w o l e b d e t a c o l m a e b
r o o lf l a n o it n e v n o c
welded
angle cleats
partial depth endplate
Introduction 290
Considering joints as separate elements
storey building
beam to column joint
Introduction 291
Characteristics of joints The initial rotational stiffness Sj,ini: The design moment resistance Mj,Rd: The design rotation capacity φCd: M
moment resistance (strength)
initial rotational stiffness rotation capacity
M
Joint Representation (General) 292
The joint representation covers all necessary actions to come from a specific joint configuration to its reproduction within the frame analysis. These actions are:
Joint
characterisation
Joint
classification
Joint
idealisation
Joint
modelling
Joint Representation (General) 293
REAL JOINT CONFIGURATION
JOINT REPRESENTATION JOINT CHARACTERISATION
ML
MS
L connection
S shear
L
S
--> COMPONENT METHOD (analytic) component identification component characterisation component assembly Joint tests FEM-Calculations Tables Software (CLASSIFICATION ) IDEALISATION realistic JOINT MODELLING
GLOBAL STRUCTURAL ANALYSIS
simplified C
Joint Characterisation 294
This chapter describes how to derive the M-φ curves, representing the necessary input data for the joint models
Joint tests
Finite Element calculations
Analytical approach
The general background for any joint model comprises three separate curves:
Left hand connection
Right hand connection
Web panel in shear
Joint Characterisation 295
Component method The procedure can be expressed in three steps:
Component identification
Component characterisation
Component assembly
Joint Characterisation 296
Component models How to divide the complex finite joint into logical parts exposed to internal forces and moments and therefore being the sources of deformations connecting zone connecting zone panel zone zones
regions
tension
4 1 5
compression shear
6 3
1 4 2
5 3 6
Joint Characterisation 297
Component models For simplifications concerning the component interplay a simplified component model has been developed Sophisticated (Innsbruck Model)
Simplified EUROCODES
Group tension (t)
S
Group tension (t) 11
C
13
12
6
8,9,10
Joint Characterisation zt
C
L
15
z
7
14
16
hj
L
zc
13
5
298
24
S
S
Component models
Group shear (S)
Group compression (c)
Group shear (S)
Group compression (c)
bj
bc
Refined component model JOINT
i
COMPONENT
1
interior steel web panel
2
concrete encasement
3
exterior steel web panel (column flange and local effects)
4
effect of concrete encasement on exterior spring
5
beam flange (local effects), contact plate, end plate
6
steel web panel incl. part of flange, fillet radius
7
stiffener in tension
8
column flange in bending (stiffened)
9
end plate in bending, beam web in tension
10 bolts in tension
GROUP Shear panel Moment Moment (only for connection connection unbalanced left right loading) M L,R MS,R M Lr,R ď Ź Load introduction
Fi,R Fi,Rd
Group tension (t)
S
Connection
S
Group tension (t) 11
rigid
semi-rigid
C
13
tension
11 reinforcement (within panel) in tension
ci,Rd weak
zt
C
wi,R
wi,Rd
L
6
15
z
12
8,9,10
7
hj
14 16
zc
L 13
5
12 slip of composite beam (due to incomplete interaction) 14 steel web panel in shear 15 steel web panel in bending 16 concrete encasement in shear
2 4
S
13 redirection of unbalanced forces
shear
Group shear (S)
S Group compression (c)
Group shear (S)
Group compression (c) bj
bc
Joint Characterisation 299
Beam-to-column joints STEEL Component model
Real joint
COMPOSITE Real joint
Component model
Joint Characterisation 300
Special joints
Real joint
Real joint
STEEL e.g. splice
COMPOSITE e.g. beam-to-beam joint
Component model 8,9,10
8,9,10
8,9,10
8,9,10
5
5
Component model 8,9,10
8,9,10
8,9,10
8,9,10
5
5
Joint Characterisation 301
ď Ź
Component assembly 1
parallel springs F
2
F1 + F 2 F2
increase of resistance
F = F1 + F2
increase of stiffness
C = C 1+ C2
minimum of deformation capacityw u= min ( wu1 , wu2 )
F1 C w u2 wu
w u1
w
Joint Characterisation 302
ď Ź
Component assembly serial springs
1
2
F F1 F2
minimum of resistance
F = min ( F 1, F 2)
decrease of stiffness
1/C = 1/C 1 + 1/C 2
increase of deformation capacity wu = w1 + w 2
C w1
w2 w1 + w 2
w
Joint Characterisation 303
Rotational stiffness S j,ini z
2
1 ci
ci represents the effective or equivalent stiffness of the region i and z is the lever arm of internal forces
Design moment resistance MRd
F
Lt,i,Rd
hi
Joint Characterisation 304
Basic components of a joint The design moment-rotation characteristic of a joint depends on the properties of its basic components.
Column web panel in shear Column web in compression Column web in tension Column flange in bending End plate in bending Beam flange and web in compression Beam web in tension Bolts in tension Longitudinal slab reinforcement in tension Contact plate in compression
Joint Classification 305
Classification of beam-to-column joints Method of global analysis
Classification of joint
Elastic
Nominally pinned
Rigid
Semi-rigid
Rigid-Plastic
Nominally pinned
Full-strength
Partial-strength
Elastic-Plastic
Nominally pinned
Rigid and full-strength Semi-rigid and partial -strength Semi-rigid and full -strength Rigid and partial-strength
Type of joint model
Simple
Continuous
Semi-Continuous
Joint Classification 306
Classification by strength
Full strength for a joint at the top of a column if:
Mj,Rd ≥ Mb,pl,Rd or:
Mj,Rd ≥ Mc,pl,Rd
Full strength for a joint within a continuous length of a column if:
or:
Mj,Rd ≥ Mb,pl,Rd Mj,Rd ≥ 2 Mc,pl,Rd
Joint Classification 307
ď Ź
Classification by strength
Nominally pinned if:
Mj,Rd ď &#x2019; 25% than the moment resistance required for a fullstrength joint
Joint Classification 308
Classification by rotational stiffness
Mj
1 2 3
EIb Lb
Zone 1: rigid if:
E Ib S j,ini 8 Lb
Zone 3: nominally pinned
E Ib S j,ini 0,5 Lb
Zone 2: semi-rigid: between case 1 and 3
uncracked flexural stiffness of composite beam span of a beam
Joint Idealisation 309
Non linear M- curve Moment Mj,Rd M j,Sd 2/3 M j,Rd Sj at Mj,Rd
Sj S j,ini
Rotation Cd
Sj
S j,ini 1,5 M j,Sd M j,Rd
Joint Idealisation 310
Possibilities for curve idealisation
non-linear
tri-linear
bi-linear M Rd
M Rd
S j,ini
S j,ini
a)
S j,ini
Cd
b)
Cd
c)
Cd
Joint representation (design provisions) 311
Basis of design: List of already covered components by prEN 1993-1-8 ď ŹCompression
region: Column web in compression Beam flange and web in compression
ď ŹTension
region: Column flange in bending Column web in tension End-plate in bending Beam web in tension Bolts in tension
Joint representation (design provisions) 312
Basis of design: List of already covered components by prEN 1993-1-8 Shear
region: Column web panel in shear
Additional basic components for composite joints: Longitudinal
slab reinforcement in tension Contact plat in compression
Joint representation (design provisions) 313
Design moment resistance Criterias for simplified calculations: The
internal forces are in equilibrium with the forces applied to the joint The
design resistance of each component is not exceeded The
deformation capacity of each component is not exceeded Compatibility
is neglected
Joint representation (design provisions) 314
Design moment resistance for a contact plate joint: FRd z FRd
Mj,Rd ď&#x20AC;˝ FRd ď&#x192;&#x2014; z
Joint representation (design provisions) 315
Determination of lever arm z:
z
z
One row of reinforcement
Two rows of reinforcement
Joint representation (design provisions) 316
Design moment resistance for a joint with steelwork connection effective in tension Ft1,Rd Ft2,Rd
z Fc,Rd
Fc,Rd= Ft1,Rd+ Ft2,Rd
Joint representation (design provisions) 317
Rotational stiffness: Sj
Ea z m
i
Ea ki z μ Sj,ini
2
1 ki
modulus of elasticity of steel stiffness coefficient for basic joint component i lever arm stiffness ratio Sj,ini / S j initial rotational stiffness of the joint, given by the expression above with μ=1,0
Joint representation (design provisions) 318
Rotational stiffness: The stiffness ratio m should be determined from the following: if: if:
Mj,Sd ≤ 2/3 Mj,Rd 2/3 Mj,Rd < if: Mj,Sd ≤ Mj,Rd
Type of connection Contact plate Bolted end-plate
μ=1
(
m 1,5 Mj,Sd / Mj,Rd Value of ψ 1,7 2,7
)
Joint representation (design provisions) 319
Initial stiffness Sj,ini: Elastic behaviour of a spring i:
Fi E k i w i Fi E ki wi
the force in the spring i; the modulus of elasticity of structural steel; the translational stiffness coefficient of the spring i the deformation of spring i.
Joint representation (design provisions) 320
Initial stiffness Sj,ini: For a joint with contact plate: k13
FRd
j
z FRd
k1
k2
M Fz F z2 E z2 S j, ini j w i F 1 1 E ki ki z
Mj
Summary 321
Field of application of the calculation procedures:
Static loading
Small normal force N in the beam (N/Npl < 0,1;where Npl is the squash load of the beam) Strong axis beam-to-column joints In the case of beam-to-beam joints and two-sided beam-to-column joints, only a slight difference in beam depth on each side is possible Steel grade: S235 to S355 H and I hot-rolled cross-sections Flush end-plates:one bolt-row in tension zone All bolt property classes as in prEN 1993-1-8 [2], it is recommended to use 8.8 or 10.9 grades of bolts One layer of longitudinal reinforcement in the slab Encased or bare column sections
Summary 322
Summary of key steps: The calculation procedures are based on the component method which requires three steps: Definition
of the active components for the studied joint;
Evaluation
of the stiffness coefficients (ki) and/or strength (FRd,i) characteristics of each individual basic component; Assembly
of the components to evaluate the stiffness (Sj) and/or resistance (Me, MRd) characteristics of the whole joint.
Summary 323
Initial stiffness: The stiffness (ki) and the design resistance (FRd,i) of each component are evaluated from analytical models. The assembly is achieved as follows: 2 Ea z S j,ini ď&#x20AC;˝
ď&#x192;Ľ
1 iď&#x20AC;˝1,n k i
where: z relevant lever arm n number of relevant components Ea elastic modulus of steel
Summary 324
Nominal stiffness: Sj
Sj
S j,ini 1,5
S j,ini 2,0
For beam-to-column joints with contact plates
For beam-to-column joints with flush end –plates
Summary 325
Plastic design moment resistance: FRd= min [FRd,i] MRd = FRd . z
Elastic design moment resistance: Me=2/3 MRd
Conclusion 326
Joints with non linear response with the three main characteristics
Initial rotational stiffness
Moment resistance
M
Rotation capacity
moment resistance (strength)
initial rotational stiffness rotation capacity
Conclusion 327
Joint representation includes the following actions for specification:
Joint characterisation
REAL JOINT CONFIGURATION
JOINT REPRESENTATION
Joint classification
Joint idealisation
Joint modelling
JOINT CHARACTERISATION
ML
MS
L connection
S shear
L
S
--> COMPONENT METHOD (analytic) component identification component characterisation component assembly Joint tests FEM-Calculations Tables Software (CLASSIFICATION ) IDEALISATION realistic JOINT MODELLING
GLOBAL STRUCTURAL ANALYSIS
simplified C
Conclusion 328
For analytical approach the COMPONENT METHOD has proved to be most appropriate: 3 steps of the component method:
Component identification
Component characterisation
Component assembly
Structural Steelwork Eurocodes 329
Advanced Composite Floor Systems
Traditional composite construction advantages 330
• off-site fabrication • speed of erection • high strength to weight ratio • long span capabilities • sustainable development recycled to produce more steel without the need for exploitation of further natural resources
• steel framed buildings are adaptable • flexibility of use • increasingly popular with clients and designers
Traditional framing systems - disadvantages 331
â&#x20AC;˘ multi-layered nature of the construction (the concrete floor is situated above the primary and secondary beams) results in relatively large structural depths â&#x20AC;˘ depths may need to be further increased by the provision of services in the ceiling space â&#x20AC;˘ provision of fire protection of the steel frame In fire, steel loses both strength and stiffness, and protection, in the form of insulating boards, intumescent paints or sprayed cementitious materials, is necessary in most cases.
Advanced composite solutions 332
Column free spacing LONG SPAN SOLUTIONS
Reduced structural and service depths INTEGRATION OF STRUCTURES AND SERVICES
Reduced structural depth SHALLOW FLOOR SYSTEMS
Costs 333
• Building frame accounts for only 10 - 15% of project cost • Choice of structural layout should be
considered in context of total project cost • Long span and shallow floor solutions can be cost effective despite increased steel content
Stub girders 334
• Suitable for spans 11.5 - 13.5m • Vierendeel-type truss • Bottom boom - column section • Top boom - concrete slab • Web - short lengths of beam: stubs
Stub girders 335
• Openings provide room for services • Require propping during construction unless temporary T- section top boom used
• Usually used in braced frames • May be modified for use in sway frames
Floor
Stub welded to bottom chord Service zone
Composite secondary beam
Floor Continuous ribs
Main ducts
Distribution ducts
Haunched beams 336
• Full strength moment resisting joint requires use of a haunch • Continuity reduces beam moments and deflections - smaller beam size • Column size may increase • Full width service zone (set by haunch depth)
Haunched beams 337
â&#x20AC;˘ Haunch length non-sway frame 5 - 7% of span sway frame 7 - 15% of span
â&#x20AC;˘ Beam size determined by moment capacity at tip of haunch Care needed: connector design haunch stability
Tapered fabricated beams 338
Application of plate girder fabrication technology Spans 15 - 20m Span:depth ratios 15 - 25
Tapered fabricated beams Castellated beam
• Shape allows flexibility for services layout • Design requires careful identification of critical sections for bending and shear • Close spacing of shear connectors at beam ends
339
Fabricated beam with straight taper
Fabricated beam with semi-taper
Fabricated beam with cranked taper
Figure 2a Types of composite floor beams for multi-storey
Composite trusses 340
• Popular for spans of 10 - 20m • Warren trusses • Services threaded through web • Large ducts - Vierendeel panel at centre
Composite trusses 341
• Unpropped behaviour during construction determines size of top boom • Member forces from elastic analysis of pinjointed truss • In composite stage - slab forms effective compression boom
Beams with web openings 342
• Uniform depth beams have reserve capacity at some locations • Small holes may be unstiffened • Larger holes require horizontal stiffeners • Customised holes are inflexible
Beams with web openings 343
â&#x20AC;˘ Castellated and cellular beams provide frequent openings for services â&#x20AC;˘ Good flexibility for future upgrading
Parallel beams 344
• Continuity reduces weight of beams • Fewer pieces reduces erection time
Service ducts Rib beam Spine beam
Figure 3 General arrangement for parallel beam approach
• Services run parallel with beams at two levels
Shallow floor systems 345
Scandanavia - building heights restricted - maximum floors in permitted height
Thor beam by ConstrucThor plc 346
Shallow floor systems 347
Precast slabs sitting on shelf-angle
Shallow floor systems 348
Compound beam with bottom plate
Shallow floor systems 349
Fabricated asymmetric beam
Shallow floor systems - Slimflor 350
1995 British Steel launched Slimflor utilises compound beams formed by the addition of a wide flat plate to the underside of universal column sections. Projecting plate forms a ledge on which concrete slab may be supported, thus permitting the steel frame and floor to occupy the same vertical space and reduce the overall thickness of the structure.
Advantages of shallow floor systems 351
Absence of downstand beams is advantageous as service runs throughout the floor area are unobstructed and greater flexibility of layout of partitions is possible.
Fire resistance 353
â&#x20AC;˘ One-hour fire resistance may be attained without the addition of any fire protection materials because the majority of the steel section is encased in concrete. â&#x20AC;˘ Longer periods can be achieved by protecting the exposed bottom flange but in many cases this is not necessary.
Composite metal decking 354
â&#x20AC;˘ Deep cold-formed metal deck permits the use of in-situ concrete acting compositely with the metal deck. â&#x20AC;˘ This reduces the dead load of the floor and removes the need to manoeuvre heavy precast sections into place.
Building frame with shallow beams 355
Metal decking 356
Steelwork ready to receive decking 357
Placing metal decking 358
Decking in place 359
Underside of floor 360
Pumping concrete 361
Asymmetric beams 362
1997 British Steel introduced new asymmetric beams engineered to fulfil dual roles 1. an efficient structural beam member to enable slim floor decks to be constructed, 2. section shape capable of providing good fire resistance without the need for fire protection. Rolled pattern on the top flange means composite action may be achieved without the need for shear stud connectors.
Rolling an ASB 363
Advantages of shallow floor systems 364
• Shallow floor depth more stories for a given height • Flexibility of service layout Inherent fire resistance Lightweight construction Speed of construction Easy fabrication and erection (low number of pieces)
Comparison of composite flooring systems
Summary of the structural designs included in the study Building A
Beam + Overall floor slab zone depth
Floor slab 365 dead load
Total steel weight per floor area
m Building height 13.0
KN/m2 Floor slab dead load 3.0
Kg/m2 Total steel weight per floor area 43.4
Building height
Summary of the structural designs included in the study STRUCTURAL FORM
Building A Slimflor with pre-cast slab Slimflor with deep metaldeck STRUCTURAL FORM (unpropped) Slimflor with deep metal deck (propped Slimflor with pre-cast slab deck, unpropped beam)
mm mm Beam + Overall floor slab zone depth 237 550 mm 305
mm 650
m 13.4
2 KN/m 2.8
2 Kg/m 51.7
295 237
650 550
13.4 13.0
2.7 3.0
42.2 43.4
Slimflor withbeams deep metaldeck Composite & composite slab (unpropped) Slimflor withconcrete deep metal Reinforced flat deck slab (propped deck, unpropped beam)
435 305
800 650
14.0 13.4
2.0 2.8
38.9 51.7
300 295
650 650
13.4 13.4
7.0 2.7
8.2 42.2
Reinforced concrete waffle slab slab Composite beams & composite
400 435
750 800
13.8 14.0
4.2 2.0
8.2 38.9
Cellular beams with composite Reinforced concrete flat slab slab
775 300
1100 650
15.2 13.4
2.0 7.0
46.6 8.2
Composite with webslab openings Reinforced beams concrete waffle
725 400
1050 750
15.0 13.8
2.0 4.2
50.2 8.2
Pre-cast concrete - hollow coreslab units Cellular beams with composite
475 775
850 1100
14.2 15.2
5.6 2.0
8.2 46.6
Pre-cast double teewith units Composite beams web openings
575 725
950 1050
14.6 15.0
3.8 2.0
8.2 50.2
Conclusions 366
• Large column free spaces often required • Simple beams may be too deep • Service integration reduces overall depth • Traditional composite construction may result in an overall structural depth which is too large • Shallow floor systems combine floor and slab in the same vertical space
Structural Steelwork Eurocodes 367
Introduction to Eurocode Structural Fire Engineering
Steel stress-strain curves at high temperatures 368
Stress (N/mm2)
Steel softens progressively from 100-200°C up. Only 23% of ambienttemperature strength remains at 700°C.
At 800°C strength reduced to 11% and at 900°C to 6%.
300 250 200
20°C 200°C 300°C 400°C 500°C
150 600°C 100
Melts at about 1500°C.
700°C
50
800°C 0
0.5
1.0 1.5 Strain (%)
2.0
Concrete stress-strain curves at high temperatures 369
Normalised stress
Concrete also loses strength and stiffness from 100°C upwards. Does not regain strength on cooling. High temperature properties depend mainly on aggregate type used.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
20°C 200°C
400°C
600°C 800°C 1000°C 1
2 Strain (%)
3
4
The fire triangle 370
Heat Fuel + Oxidant = Combustion products CH4 + O2 = CO2 + 2H20 Reaction occurs when Oxygen/fuel mixture hot enough Fuel
Oxygen
Stages of a natural fire - and the standard fire test curve 371
Temperature
Post-Flashover 1000-1200°C
Pre-Flashover Flashover
Natural fire curve
ISO834 standard fire curve Time Ignition - Smouldering
Heating
Cooling â&#x20AC;Ś.
The EC1 (ISO834) standard fire curve 372
Gas Temperature (°C) 1000
945
900
700
842 781 739 675
600
576
800
500
20 345 log( 8t 1 ) { t in min utes }
400 300 200 100 0
0
600
1200
1800 2400 Time (sec)
3000
3600
Different EC1 time-temperature curves 373
Gas Temperature (°C)
Fire resistance times based 1200 on standard furnace tests NOT on survival in real 1000 fires.
Hydrocarbon Fire
Standard Fire
800
EC1 Parametric Fire 600 temperature-time curves. Based on fire load and 400 compartment properties (<500m2). Only allowed with calculation models. 200
External Fire Typical EC1 Parametric fire curve
0
1200
2400 Time (sec)
3600
Time-equivalence
ď Ź
Matches times to given temperature in a natural fire and in Standard Fire.
Used to rate fire severity or element performance relative to furnace test. Fire severity time equivalent
Fire resistance time equivalent
Natural fire
Standard fire
Time Temperature
ď Ź
Load-bearing resistance
374
Compartment Element Time
Furnace tests on structural elements 375
Fire Testing Load kept constant, fire temperature increased using Standard Fire curve.
Maximum deflection criterion for fire resistance of beams. Load capacity criterion for fire resistance of columns.
Problems Limited range of spans feasible, simply supported beams only.
Effects of continuity ignored. Beams fail by “run-away”.
Restraint to thermal expansion by surrounding structure ignored.
Standard fire resistance furnace test 376
Deflection (mm) 300
200
100
0
1200 2400 Time (sec)
3600
Standard fire resistance furnace test 377
Deflection (mm) 300 Span2/400d If rate < span2/9000d
200
Span/30 100 Standard Fire 0
1200 2400 Time (sec)
3600
Structural fire protection 378
Passive Protection
Insulating Board Gypsum, Mineral fibre, Vermiculite. Easy to apply, aesthetically acceptable. Difficulties with complex details.
Cementitious Sprays Mineral fibre or vermiculite in cement binder. Cheap to apply, but messy; clean-up may be expensive. Poor aesthetics; normally used behind suspended ceilings.
Intumescent Paints Decorative finish under normal conditions. Expands on heating to produce insulating layer. Can now be done off-site.
Inherent fire protection to steel beams 379
Downstand Beam
“Slim-floor” Systems
Shelf-angle Beam
Composite sections 380
Passive Protection – Composite sections
Traditional downstand beam top flange upper face totally shielded by the slab
Downstand Beam
Composite sections 381
Passive Protection – Composite sections
Beams with concrete encasement Have high fire resistance (up to 180 minutes). Involve complicated construction of joints. Require formwork.
Encased Beam
Composite sections 382
Passive Protection – Composite sections
Steel beams with partial concrete encasement
Concrete between flanges reduces the rate of heating of the profile's web and upper flange. Concrete between flanges contributes to the load-bearing resistance. The beam can be fabricated in the workshop without the use of formwork. Simple construction of joints.
Partially Encased Beam
Load reduction factor in fire 383
Either …..
Or more usefully…..
fi
E fi .d .t
fi
E fi .d .t
Rd Ed
GAGk 1.1Qk .1 fi G Gk Q .1Qk .1
Relative to ambient-temperature design resistance
Relative to ambient-temperature design load (more conservative)
Establishing Fire Resistance: Strategies 384
Eurocodes allow fire resistance to be established in any of 3 “domains”:
Time:
tfi.d > tfi.requ
Load resistance: Rfi.d.t > Efi.d.t Temperature:
• Usually only directly feasible using advanced calculation models.
cr.d > d
• Feasible by hand calculation. Find reduced resistance at design temperature. • Most usual simple EC3 method. Find critical temperature for loading, compare with design temperature.
Material properties 385
Steel
Mechanical
Concrete
Mechanical
(effective yield strength,
(compressive strength,
elastic modulus, ... )
secant modulus, ... )
Thermal
Thermal
(thermal expansion,
(thermal expansion,
thermal conductivity,
thermal conductivity,
specific heat)
specific heat)
Steel stress-strain curves at high temperatures 386
Stress (N/mm2)
Strength/stiffness reduction factors for elastic modulus and yield strength (2% strain).
300 250 200
Elastic modulus at 600°C reduced by about 70%.
20°C 200°C 300°C 400°C 500°C
150 600°C 100
Yield strength at 600°C reduced by over 50%.
700°C
50
800°C 0
0.5
1.0 1.5 Strain (%)
2.0
Degradation of steel strength and stiffness 387
% of normal value • Strength and stiffness reductions very similar for S235, S275, S355 structural steels and hotrolled reinforcing bars. (SS)
• Cold-worked reinforcing bars S500 deteriorate more rapidly. (Rft)
100 Rft
Effective yield strength (at 2% strain)
80
SS
60 40 20
0
Rft
SS
Elastic modulus
300
600
900
Temperature (°C)
1200
Degradation of concrete strength and stiffness 388
Strength reduction factors
Accurate for normal density concrete with siliceous aggregates.
Strength (% of normal)
Strain (%) 6
Lightweight Concrete
100
5 4
Conservative for normal density concrete with calcareous aggregates,.
Strain at maximum strength 50 Normal-weight Concrete
3
Conservative for lightweight concretes. All types treated the same.
2 1
0
200
400
600 800 1000 1200 Temperature (°C)
Concrete strength in heating and cooling 389
25
Stress-strain relationship at ambient temperature
Stress-strain relationship in heating phase (700C)
15 Stress-strain relationship in heating phase (400C)
Stress-strain relationship in cooling from 700°C (at 400C)
5 0,01
0,02
Stress-strain relationship after cooling from 700°C (at 20C)
0,03
Thermal expansion of steel and concrete 390
Expansion Coeff /°C (x 10-6) • Steel thermal expansion stops during crystal structrure change in the 700-800°C range.
4,5 4,0
3,5
Normal-weight concrete
3,0
• Concrete unlikely to reach 700°C in time of a building fire.
2,5
• Light-weight concrete treated as having uniform thermal expansion coefficient.
1,0
2,0
Steel
1,5 0,5
Lightweight concrete
0 100 200 300 400 500 600 700 800 900 Temperature (°C)
Other steel thermal properties 391
Thermal conductivity (W/m°K)
Specific Heat (J/kg°K)
60
5000
50
a=45W/m°K (EC3 simple calculation model)
ca=600J/kg°K (EC3 simple calculation model)
4000
40 30
Steel
2000
20
Steel 1000
10 0
3000
200 400 600 800 1000 1200 Temperature (°C)
0
200 400
600 800 1000 1200 Temperature (°C)
Other concrete thermal properties 392
May assume constant value for NC:
May assume constant value for NC: 1200
1,60 W/m.K
3 NC
cc* 1000 J/kg.K
NC
1000
2
800
1 LC
400 200
600
1000 °C
Thermal conductivity c (W/m.K)
LC 200
600
Specific heat cc (J/kg.K)
1000 °C
Thermal analysis 393
Thermal analysis:
â&#x20AC;˘ both EC3 Part 1.2 and EC4 Part 1.2
â&#x20AC;˘ unprotected and protected steel beams Lower and upper flanges
proper calculation of temperatures
!
Considerably different temperatures Temperature
Temperature increase of unprotected steel 394
Temperature increase in time step t:
a .t
Fire temperature
Am hnet .d t ca a V 1
Steel temperature
Steel Heat flux hnet.d has 2 parts: Radiation:
(
hnet.r 5,67 x10 res ( r 273) ( m 273) 8
Convection:
hnet ,c c ( g m )
4
4
)
Section factor Am/V unprotected steel members 395
90%
!
b
h
perimeter c/s area
exposed perimeter c/s area
2(b+h) c/s area
Temperature increase of protected steel 396
• Some heat stored in protection layer.
Fire temperature
Steel temperature
• Heat stored in protection layer relative to heat stored in steel
cp p ca a
dp
Ap
Steel
V
• Temperature rise of steel in time
Protection
dp
increment t
a .t
p / d p Ap 1 ( g .t a .t )t (e / 10 1) g .t ca a V 1 / 3
Section factor Am/V inherently protected systems 397
exposed perimeter
exposed plate
exposed flange
Total c/s area
Total c/s area
Total c/s area
Section factor Ap/V protected steel members 398
90%
!
b
h
Steel perimeter steel c/s area
inner perimeter of board steel c/s area
2(b+h) c/s area
Structural Steelwork Eurocodes 399
Fire Engineering Design of Composite Structures
Validity of EC 4 Part 1.2 Slabs 400
COMPOSITE SLABS:
Validity of EC 4 Part 1.2 Beams 401
COMPOSITE BEAMS:
Validity of EC 4 Part 1.2 Columns 402
COMPOSITE COLUMNS:
Structural fire design Criteria 403
CRITERIA: • Integrity criterion “E”
• Insulation criterion “I”
• Load bearing criterion “R”
Structural fire design Design methods 404
ENV 1994-1-2: •
Tabular data (recognized design solutions for specific types of structural members)
•
Simple calculation models (for specific types of structural members)
•
Advanced calculation models (global structure, parts of the structure, structural members)
Simple calculation models Composite slabs – Criterion “E“ 405
UNPROTECTED COMPOSITE SLABS
1. INTEGRITY CRITERION “E” •
satisfied automatically (if designed according to EC4-1-1)
Composite slabs - Criterion “I“ 406
2. INSULATION CRITERION “I”
l1 l2 l1 l3
heff = h1 + 0,5h2
l1 l2 1 + 0,75 l l 1 3
heff = h1
(for h2 / h1 1,5 )
(for h2 / h1 > 1,5 )
Screed
h3 h1 h2
heff
l2 l1
Concrete Steel sheet
l1 l3
l2
l3
Composite slabs - Criterion “I“ 407
heff heff,min : Standard Fire Resistance
Minimum effective thickness
R30
60 – h3
R90
100 – h3
R180
150 – h3
( Screed thickness = h3
Max 20 mm taken into account)
For lightweight concrete values 10 % lower
Composite slabs - Criterion “R“ 408
3. LOAD BEARING CRITERION “R”
Assumptions: • The steel decking is not taken into account • Evaluation of the load-bearing capacity is based on the analysis - sufficient rotational capacity - tensile reinforcement with sufficient deformation capacity - an adequate reinforcement ratio
plastic global
Composite slabs - Criterion â&#x20AC;&#x153;Râ&#x20AC;&#x153; Sagging moment resistance 409
SAGGING MOMENT RESISTANCE: 1. Steel decking + Concrete in tension
2. Concrete in compression without reduction of strength
NEGLECTED
The sagging moment resistance depends on the amount of tensile reinforcement and its temperature.
Composite slabs - Criterion “R“ Sagging moment resistance 410
Slab Rebar u1 u3 u2
u2
u1
Steel sheet
u3
1 1 1 1 z u1 u2 u3 Standard Fire Resistance
Temperature of the reinforcement [°C]
R60
s = 1175 - 350 z 810°C for (z 3,3)
R120
s = 1370 - 350 z 930°C for (z 3,8)
Composite slabs - Criterion “R“ Hogging moment resistance 411
HOGGING MOMENT RESISTANCE: Concrete in compression is on the exposed side
Reduced strength:
• Integration over the depth of the ribs • replacing the ribbed slab by an equivalent slab of uniform thickness heff
[mm] 100
0 100
705 [°C]
Protected composite slabs 412
Protected composite slabs
Composite slab with insulating coating
Composite slab with suspended ceiling
If temperature of the steel sheeting 350 °C
The load-bearing criterion “R” is fulfilled automatically
Composite beams - (A) 413
Composite Beams Including Steel Sections with no Concrete Encasement
Thermal analysis (Lecture 11a)
Mechanical analysis
Composite beams (A) Mechanical analysis 414
A. MECHANICAL ANALYSIS •
The Critical Temperature Method
- simply supported beams - hot- rolled sections
•
uniform temperature over the depth
- steel section: h 500 mm
The Bending Moment Resistance Method
- steel section: h 500 mm
- concrete slab: hc 120 mm
and / or - concrete slab: hc 120 mm
!
Composite beams (A) Critical temperature method 415
Temperature
ISO 834
fi,t
unprotected section
crit fi,t
protected section
treq
fi,t crit
!
Time
Composite beams (A) Critical temperature method 416
crit fi,t
Efi ,d ,t Rd
f(fi,t)
(load level)
the ultimate limit state is reached when Rfi,d,t decreases to the level Efi,d,t
fi,t
Rfi ,d ,t Rd
Composite beams (A) Critical temperature method 417
High temperature of steel section ... causes the neutral axis position to be high ... only a small part of the slab is in compression ...
The bending moment resistance in the fire situation is influenced mainly by the steel strength.
fi ,t
Rfi ,d ,t fa max,cr / M ,fi ,a 1,0 Rd fay ,20C / M ,a 1,1
0 ,9 fi ,t
fa max,cr fay ,20C
crit fi,t
!
Composite beams – (A) Moment Resistance Method 418
The Bending Moment Resistance Method
• Steel section Class 1 or 2
Simple plastic theory
• Sufficient rotational capacity of the concrete slab (Must fulfil EC2 Part 1.2 requirements)
f w f
Composite beams (A) Moment Resistance Method 419
Neutral axis position
Equilibrium of tensile force T and compressive force F
Depth of concrete in compression Sagging moment resistance:
M
fi,Rd + hu
ď&#x20AC;˝ T (yF - yT ) -
+
F T
yT
yF
Composite beams (A) Shear resistance 420
Composite beams
Slab and the steel section act as a single structural member
Shear connectors
Shear resistance
Sufficient strength and stiffness to resist shear
Pfi,Rd PRd
d 2 0,8 fu 4 k max, kmax, M ,fi ,v
Pfi,Rd PRd
0,29d 2 k c, kc, fck Ecm M ,fi ,v
= min
Composite beams (B) 421
Steel beam with partial concrete encasement
min 90% â&#x20AC;˘ Simply supported / Continuous
â&#x20AC;˘ Three-sided exposure Restrictions (example): Restricted dimensions
Fire Resistance Class R30
R90
Minimum slab thickness hc [mm]
60
100
Minimum profile height h and width bc [mm]
120
170
17500
35000
Minimum area h .bc [mm2]
Composite beams (B) Thermal analysis 422
THERMAL ANALYSIS • Lower steel flange – heated directly • Other steel parts – protected by the
complicated heating concrete
estimation of the temperatures of the individual parts of the section by simple calculation
Composite beams (B) Reduction of the cross-section 423
Lower steel flange and web, rebars between flanges
Full section and reduced strength
Concrete infill, the lower parts hc,fi of the concrete slab, the ends bfi of the upper steel flange
Full strength and reduced section
Composite beams (B) Mechanical analysis 424
Checking of
Simply supported beams
Continuous beams
M fi ,Sd M fi ,Rd
M fi ,Sd M fi ,Rd
M fi ,Sd M fi ,Rd
Composite beams (B) Mechanical analysis 425
Simply supported beam Continuous bar
IN FIRE
Studs
Effective transmission of the compression force through the steel connection Gap
Continuous beam in the fire case
Sections with concrete infill
Composite beams (B) Sagging moment resistance 426
Mfi,Rd+
Estimation of the reduced section
Calculation of the sagging moment resistance
(-)
(+)
+
Composite beams (B) Sagging moment resistance 427
Estimation of the reduced section • Only the part in compression not influenced by temperature
Concrete slab
hc,fi
• Compressive concrete strength fc,20°C/M,fi,c • Reduced thickness hc,fi varies with the fire resistance class
bfi
Upper flange of steel section
• Heated edges bfi are not taken into account • Strength fay,20°C/M,fi,a
• bfi is related to the fire resistance class
Composite beams (B) Sagging moment resistance 428
Web of steel section • Assumed to remain at 20°C
Upper part hh
• Strength fay,20°C/M,fi,a hh hl
Lower part h
• Temperature changes linearly from 20°C at its top edge to the temperature of the lower flange at its bottom edge • The height h varies with the fire resistance class
Composite beams (B) Sagging moment resistance 429
Lower flange of steel section
• Uniform temperature distribution • Full area • Yield point reduced by the factor ka depending on the fire resistance class
us ui
Reinforcing bars
• Temperature depends on the distance from the lower flange ui and on the concrete cover us • The reduction factor kr is calculated by the empirical formula
Concrete between flanges
• Not included in the calculation of sagging moment resistance (but must resist the vertical shear by itself)
Composite beams (B) Sagging moment resistance 430
Definition of the neutral axis
• Plastic distribution of stresses
Calculation of Mfi,Rd+
• Summation of the contributions of each of the stress blocks shown
• Equilibrium of tensile and compressive resultants
(-)
(+)
-
Mfi ,Sd
Mfi ,Rd
!
+
Composite beams (B) Hogging moment resistance 431
Mfi,RdEstimation of the reduced section
Calculation of the hogging moment resistance
Composite beams – B Cross-section reduction 432
Estimation of the reduced section • Concrete in tension is excluded from the calculation
Concrete slab and reinforcement beff = 3.b
• Tensile reinforcement lying in the effective area (beff=3b) is taken into account
uh ul
bfi
Upper flange of steel section
• The reduction factor ks depends on the distance u
• The same rules as for sagging moment resistance • Simply supported beam - the upper flange should not be taken into account if it is in tension
Composite beams – B Cross-section reduction 433
Concrete between the flanges
• Full compressive strength • Reduced cross-section (hfi and bfi depend on the fire resistance class)
bc,fi hfi Reinforcing bars
• The same rules as for sagging moment resistance • Web is assumed to transmit the shear force (neglected when calculating the hogging bending moment resistance)
Steel web and lower flange
• Compressed lower flange should be ignored
Composite beams – B Hogging moment resistance 434
Definition of the neutral axis
• Plastic distribution of stresses
Calculation of Mfi,Rd -
• Summation of the contributions of each of the stress blocks shown
Mfi,Sd Mfi,Rd
• Equilibrium of tensile and compressive resultants
!
Slim-floor beams 435
ADVANTAGES:
Low depth of the floor structure
Good inherent fire resistance (up to 60 min.)
The fire resistance is not directly covered in simplified methods within EC4 Part 1.2
!
Slim-floor beams General principles 436
Temperature distribution
• Two-dimensional heat transfer model • Thermal properties of materials taken from EC4 Part 1.2 • Heat flux should be determined by considering thermal radiation and convection
Fire resistance
• The moment capacity method • Divide the section into several components:
– Plate and/or the bottom flange – The lower web, the upper web
– The upper flange – Reinforcing bars
– The concrete slab (ignored in tension)
Composite columns 437
METHODS FOR
Steel sections with partial concrete encasement Concrete-filled circular and square hollow sections
Buckling Resistance in Fire
Nfi ,Rd ,z z . Nfi ,pl .Rd
• Buckling about the minor axis z • Buckling curve (c)
Composite columns 438
Validity of EC4 Part 1.2 rules
only for braced frames Bracing system
Fire limited to a single storey
The fire-affected columns fully connected to the colder columns below and above
lfi=0,7L
lfi
lfi=0,5L
Steel section with partial concrete encasement 439
Restrictions:
Buckling length l 13,5 b
Depth of cross-section h is between 230mm and 1100mm Width of cross-section b is between 230mm and 500mm Minimum h and b for R90 and R120 is 300mm Percentage of reinforcing steel is between 1% and 6%
Standard fire resistance period 120min
Partial concrete encasement Division of the cross-section 440
Division into zones: • Flanges of the steel section • Web of the steel section
• Reinforcing bars
• Concrete infill
Partial concrete encasement Division of the cross-section 441
Flanges of the steel section â&#x20AC;˘ Uniform temperature distribution (estimated for the required fire resistance class)
â&#x20AC;˘ Reduced strength and modulus of elasticity is a function of temperature
Partial concrete encasement Division of the cross-section 442
Web of the steel section • Outer parts have a considerably higher temperature high thermal gradient occurs
• The outer parts hw,fi are ignored
• Reduced strength and full modulus of elasticity
hw,fi
Partial concrete encasement Division of the cross-section 443
Concrete infill • Outer parts have a considerably higher temperature high thermal gradient occurs
bc,fi
• The outer parts bc,fi are ignored • Reduced strength and modulus of elasticity varies with the fire resistance class and section factor
bc,fi
Partial concrete encasement: Division of the cross-section 444
Reinforcing bars â&#x20AC;˘ Uniform temperature distribution (estimated for the required fire resistance class) â&#x20AC;˘ Reduction coefficients for strength and modulus of elasticity depend on the fire resistance class and distance u
u ď&#x20AC;˝ u1.u2 u2
u1
Composite columns: Design procedure 445
Plastic resistance to axial compression
Effective flexural stiffness
Determination of critical length
Euler critical buckling load
Non-dimensional slenderness ratio
Buckling resistance
Verification
Composite columns: Design procedure 446
• steel • concrete
Plastic resistance to axial compression
• reinforcement Nfi,pl.Rd = Nfi,pl.Rd,a + Nfi,pl.Rd,c + Nfi,pl.Rd,s
(Aa, famax, ) M,fi,a j
(Ac, fc, ) M,fi,c m
(As, fsmax, ) M,fi,s k
Composite columns: Design procedure 447
• steel • concrete
Effective flexural stiffness
• reinforcement
(EI)fi,eff = (EaIa)fi,eff + (EcIc)fi,eff + (EsIs)fi,eff
( E I ) j
a,
a, θ a, θ
( E I ) m
c,
c, θ c, θ
( E I ) k
s,
s, θ s, θ
Composite columns: Design procedure 448
Determination of critical length
l = 0,5 lcr or 0,7 lcr (as stated before)
2 (EI )fi ,eff ,z
Euler critical buckling load
Nfi ,cr ,z
Non-dimensional slenderness ratio
Nfi ,pl ,R Nfi ,cr ,z
l2
z for buckling curve “c”
Composite columns: Design procedure 449
Buckling resistance
Verification
Nfi,Rd,z = z Nfi,pl,Rd Nfi,Sd Nfi,Rd,z
!
(fi NSd Nfi,Rd,z)
Eccentricity of loading - design buckling load for The application point should remain inside the composite section of the column
Nfi ,Rd, Nfi ,Rd
NRd, NRd
Normal temperature design
buckling resistance for eccentric loading axial buckling resistance
Composite columns: Unprotected concrete-filled hollow sections 450
Filling the steel hollow sections with concrete
• Increases load-bearing capacity
• May allow reduction of the section size • Rapid erection without requiring formwork
increases the local buckling resistance of the steel section
• High inherent fire resistance without additional fire protection
Confines the concrete laterally Protects it from direct fire exposure and prevents spalling (slower decrease of mechanical properties of concrete)
Composite columns: Unprotected concrete-filled hollow sections 451
Behaviour during the fire: Later stages of fire exposure
First stages of fire exposure
• The strength of steel begins to degrade rapidly (high temperature)
• Steel part expands more rapidly than the concrete • The heating of the core is relatively slow
The concrete part takes over the load-carrying rôle the steel section carries most of the load FAILURE buckling
compression
Composite columns: Unprotected concrete filled hollow sections 452
RESTRICTIONS:
High temperatures Free moisture-content of the concrete and chemically bonded water of crystallisation is driven out Openings at both the top and bottom of each storey
circular and square hollow sections only Buckling length l 4,5 m
Depth b or diameter d of cross-section is between 140 and 400 mm Concrete grade is either C20/25 or C40/50
Percentage of reinforcing steel is between 0% and 5% Standard fire resistance period 120 min
Composite columns: Unprotected concrete filled hollow sections 453
Analysis
Calculation of temperatures over the cross-section
Calculation of the buckling resistance in fire
• The temperature of the steel wall is homogeneous • There is no thermal resistance between the steel wall and the concrete • The temperature of the reinforcing bars is equal to the temperature of the concrete surrounding them • There is no longitudinal thermal gradient along the column
Composite columns: Unprotected concrete filled hollow sections 454
The net heat flux transmitted to the concrete core:
a h net ,d acae t
Finite difference or
The heat transfer in the concrete core:
c c c c, c, cc,c t y y z z
Finite element methods
Composite columns: Unprotected concrete filled hollow sections 455
The design buckling resistance in fire
As for concrete-encased sections (with some differencies)
N fi ,Rd N fi ,pl ,Rd
The plastic resistance
Sum of the plastic resistances of all components (the wall of the steel section, reinforcing bars and the concrete core)
Nfi,pl.Rd Nfi,pl.Rd, a Nfi,pl.Rd,c Nfi,pl.Rd, s
Composite columns: Unprotected concrete-filled hollow sections 456
Euler critical load:
Nfi ,cr
2 Ea,θ, Ia+E c,θ, Ic+E s,θ, Is l2
Rebars
Rebars Steel wall
Steel wall
Concrete
Concrete temperature [°C]
Degradation of strength ...
temperature [°C]
Degradation of tangent stiffness ...
… for the constituent parts of concrete-filled hollow sections
Composite columns: Unprotected concrete-filled hollow sections 457
The design buckling resistance in fire
Available in tabular or
graphical form
Composite columns: Unprotected concrete-filled hollow sections 458
Eccentricity of loading
Equivalent axial load Nequ:
Correction coefficient for the percentage of reinforcement
Nequ
Nfi ,Sd
s
Coefficient which takes account of the eccentricity of loading
Protected concrete-filled hollow sections
The load-bearing criterion is fulfilled provided that the temperature of the steel wall remains below 350°C
Tabular data 459
Recognized design solutions
• For specific types of structural members • For some special cases under standard fire conditions • For braced frames Restrictions • Neither the boundary conditions nor the internal forces at the ends of members change during the fire • The loading actions are not time-dependent • The fire resistance depends on the load level fi,t
Tabular data 460
Simply supported beams
• Composite beams comprising a steel beam with partial concrete encasement • Encased steel beams, for which the concrete has only an insulating function Columns • Composite columns comprising totally encased steel sections • Composite columns comprising partially encased steel sections • Composite columns comprising concrete-filled hollow sections
Tabular data: Example 461
Condition for application: Slab: hc 120 mm
beff
beff 5 m
hc Ac
Steel section:
As
Af=b x ef
h u1 ef ew
u2 b
Standard Fire Resistance
b / ew 15
ef / e w 2 Additional reinforcement area, related to total area between the flanges: As/(Ac + As) 5%
R30 R60 R90 R120 R180
1 Minimum cross-sectional dimensions for load level fi,t = 0,3 Min b [mm] and additional reinforcement As in relation to the area of flange As / Af 1.1 h 0,9 min b (HEB 400: h = 1,3b; b = 300) 1.2 h 1,5 min b 1.3 h 2,0 min b
70/0,0 100/0,0 170/0,0 200/0,0 260/0,0 60/0,0 100/0,0 150/0,0 180/0,0 240/0,0 60/0,0 100/0,0 150/0,0 180/0,0 240/0,0
Advanced calculation models 462
Advanced calculation models
Realistic analysis of structures exposed to fire
Models: • Individual members • Subassemblies • Entire structures
• Thermal response model (the development and distribution of temperature within a structural element) • Mechanical response model (the mechanical behaviour of the structure or of any part of it)
Worked Examples: Composite beams 463
1. Composite beam comprising concrete slab and: a)
An unprotected steel section
b) A steel section protected by sprayed material
c)
A steel section protected by gypsum boarding.
2. Composite beam comprising concrete slab and a partially encased steel section
Worked Example 1: Composite beams 464
Composite beam comprising a concrete slab and a steel section:
Steel: Concrete slab: Standard fire resistance:
IPE 400 hc = 120 mm;
beff = 1000 mm
R90 Distance between beams = 6,0 m
7,0 m
Characteristic values
Load factors
Design values
Dead load
3,6 kN/m2
3,6 . 6,0 = 21,6 kN/m
ď §G = 1,35
29,16 kN/m
Imposed load
4,5 kN/m2
4,5 . 6,0 = 27,0 kN/m
ď §Q1 = 1,5
40,50 kN/m
Total
8,1 kN/m2
48,6 kN/m
69,66 kN/m
Worked Example 1: Composite beams 465
Section Properties: Steel grade:
S235
Concrete grade:
C20/25
1000 120
Steel area:
8450 mm2
Concrete area:
120x103
Mu,pl = 425,18 kNm
8,6
mm2
13,5 400
180
Worked Example 1: Composite beams 466
h < 500 mm [= 400] hc 120 mm [= 120] simply supported beam
Critical temperature model
Loading in the fire situation:
GA .Gk 1,1.Qk,1 2,i .Qk,i Ad qfi 1,0.21,6 0,7.27,0 40,5kN/m Mfi ,Sd
1 40,5.72 248,06kNm 8
Worked Example 1: Composite beams 467
The load level:
fi ,t
Efi ,d ,t Mfi ,Sd 248 ,06 0 ,583 Rd Mu ,pl 425 ,18
Calculation of the critical temperature:
0 ,9fi ,t
k max,
fa max,cr fay ,20C
123 ,4 0 ,525 235
fa max,cr 0 ,9fi ,t fay ,20C 0 ,9.0 ,583 .235 123 ,4 MPa
crit = 582°C
Worked Example 1: Composite beams 468
Unprotected steel section
Temperature increase:
a.t
1
Am hnet.d t ca a V
ca = specific heat of steel [600 J/kgK] a = density of steel
[7850kg/m3]
hnet,d = design value of net heat flux per unit area [7.1.3.1]
[f = 0,8; m = 0,625]
Am = the “Section Factor” V
Worked Example 1: Composite beams 469
Estimation of section factor:
Am 1,47 m 2 / m V 8450 .10 6 m 3 / m Am ,3 Am 0 ,180
1000 120
1,470 0 ,180 1,290 m 2 / m 8,6
Am ,3 1,290 1 153 m V 8450 .10 6
13,5 400
180
Worked Example 1: Composite beams 470
Unprotected steel section
Temperature increase:
a.t
1
Am hnet.d t ca a V
t = 5 s
ca
= specific heat of steel [600 J/kgK]
a
= density of steel
[7850kg/m3]
hnet,d = design value of net heat flux per unit area [7.1.3.1] [f = 0,8; m = 0,625]
Am = the “Section Factor” [153 m-1] V
tfi,Sd = 13,4 min < 90 min
Worked Example 1: Composite beams 471
Protected – sprayed protection
Temperature increase:
a.t
p / d p Ap 1 ( g .t a.t )t (e / 10 1) g .t ca a V 1 / 3
cp = [1200 J/kgK]
p
p = [430kg/m3]
a.t, g.t = temperatures of steel and furnace gas at time t
dp = [0,025 mm] p = [0,174 W/mK]
t = 30s
g.t
= [0,419]
= increase of gas temperature during the time step t
Am = the “Section Factor” [153 m-1] V
Worked Example 1: Composite beams 472
Protected – sprayed protection
Sprayed protection is applied directly to the surface of the steel section
1000 120
8,6
A 153m1 m V V
Ap
400
180
tfi,Sd = 90,1 min > 90 min
13,5
Worked Example 1: Composite beams 473
Protection – gypsum boarding
Temperature increase:
a.t
p / d p Ap 1 ( g .t a.t )t (e / 10 1) g .t ca a V 1 / 3
cp = [1700 J/kgK]
p
p = [800 kg/m3]
a.t, g.t = temperatures of steel and furnace gas at time t
dp = [0,020 mm] p = [0, 200 W/mK]
t = 30s
g.t
= [0,7369]
= increase of gas temperature during the time step t
Am = the “Section Factor” [153 m-1] V
Worked Example 1: Composite beams 474
Protection – gypsum boarding
Boxed protection:
1000
V 8450.10 6 m 3 / m
120
Am ,3 0 ,180 2.0 ,400 0 ,980m 2 / m Am ,3 V
0 ,980 8450.10
6
116m
13,5 400
1
180
tfi,Sd = 90,5 min > 90 min
Worked Example 2: Composite beams 475
Composite beam comprising a concrete slab and a partially encased steel section Steel: Concrete slab: Stand. fire resist.: R90
IPE 400 hc = 120 mm; beff = 1000 mm Distance betwwen beams = 6,0 m
8,0 m
Characteristic values
Load factors
Design values
Dead load
3,6 kN/m2
3,6 . 6,0 = 21,6 kN/m
ď §G = 1,35
29,16 kN/m
Imposed load
4,2 kN/m2
4,2 . 6,0 = 25,2 kN/m
ď §Q1 = 1,5
37,80 kN/m
Total
7,8 kN/m2
46,8 kN/m
66,96 kN/m
Worked Example 2: Composite beams 476
Section characteristics:
Steel grade:
S235
Concrete grade:
C20/25
Steel area:
8450 mm2
1000 120 13,5
Concrete slab area: 120 . 103 mm2
8,6
Additional reinforcement area: 1256,6 mm2 180
Mu,pl = 625,38 kNm
400
Worked Example 2: Composite beams 477
Loading in the fire situation:
GA .Gk 1,1 .Qk ,1 2 ,i .Qk ,i Ad
qfi 1,0.21,6 0,7.25,2 39,4kN/m Mfi ,Sd
1 39,24.82 313,92kNm 8
Mfi ,Sd Mfi ,Rd 313 ,92 kNm Mfi ,Rd
!
Worked Example 2: Composite beams 478
Reduction of the cross-section for R90
Reduction of the concrete thickness for R90: hc,fi = 30 mm
37 = b fi
hc,fi=30
hc,h = 90 mm Reduction of the width of the upper flange:
bfi (ef / 2) 30 (b bc ) / 2
bfi (13,5 / 2) 30 (180 180 ) / 2 bfi 36,75mm
2 1
Worked Example 2: Composite beams 479
Reduction of the cross-section for R90
Upper and lower parts of web for R90:
hl a1 / bc a2 ew / (bc h )
h 400 2 ,2 2 bc 180
286,3=h h 2 1
a1 14000 mm2 a2 75000 mm hl,min 40 mm
2
86,7=h
hl 14000 / 180 75000 8 ,6 / (180 400 ) hl 86 ,74 mm
hh 286 ,26 mm
Worked Example 2: Composite beams 480
Reduction of the cross-section for R90
Strength Reduction Coefficient for the lower flange for R90: k a 0,12 17 / bc h / (38bc ) a0
k a 0,12 17 / 180 400 / (38 180 ) 0,943 0,06 k a 0,079 0,12
a0 0,018ef 0,7 a0 0,018 13,5 0,7 a0 0,943
2 1 120=u1
Strength Reduction Coefficient for the reinforcement for R90:
60 =u s1,2
1 (ua3 a4 )a5 u kr 1 1 Am
V u(1 ) 29 ,43 mm
ui
usi
1 bc ew usi
u( 2 ) 31,72 mm
170=u2
Worked Example 2: Composite beams 481
Reduction of the cross-section for R90
Estimation of the section factor:
Am 2h bc 2 400 180
400
Am 980 mm V h bc 72000 mm 2 180
Worked Example 2: Composite beams 482
Reduction of the cross-section for R90
Strength Reduction Coefficient for the reinforcement for R90: kr
(ua3 a4 )a5 Am V
a3=0,026
u
1 1 1 1 ui usi bc ew usi
a4=-0,154
2 1
a5=0,090
120=u1
u(1) 29,43mm
k r (1)
u(2) 31,72mm
(29,43 0,026 0,154 ) 0,090 0,471 980 72000
170=u2
60 =u s1,2
k r ( 2)
(31,72 0,026 0,154 ) 0,090 0,517 980 72000
Worked Example 2: Composite beams 483
Calculation of position of the neutral axis
FH FH
FH 0 :
+ hc,h
+
-
x N f,1 N c,h c,h
Assumption: the neutral axis position is in the concrete slab (hc,h):
N w,h
h
Nr,2
lin.
N w,h const. N w,h
N r,1
N f,2
. lin . FH Nf 1 Nw ,h Nwconst N ,h w ,h h
l
l
Nf 2 Nr 1 Nr 2 1375 ,24 kN
1375,24 103 x 80,9mm 90mm 17000
FH 0 ,85 fc ,20 beff x 0 ,85 20 1000 x 17000 x
Worked Example 2: Composite beams 484
Bending moment resistance:
Mu,fi,ď ącr = ď &#x201C;My = 315,59kNm > 313,92 kNm
!
The beam has a standard fire resistance in excess of 90 min +
-
+
81 N f,1 N w,h
h
439
N c,h c,h
Nr,2
lin.
N w,h const. N w,h
N f,2
N r,1
Worked Example 2: Composite columns 485
Composite column composed of a partially encased HEB 600 steel section
Simple calculation model
Worked Example 3: Composite columns 486
• Seven-storey building frame structure, storey height: • Spacing of frames: • Span of the main beam: 8,0 m Characteristic values
4,0 m 6,0 m
Load factors
Design values
Dead load
4,65 kN/m 2
4,65 . 6,0 . 8,0 . 7 = 1562,4 kN
G = 1,35
2109,24 kN
Imposed load
6,10 kN/m
2
6,10 . 6,0 . 8,0 . 7 = 2049,6 kN
Q1 = 1,5
3074,40 kN
Total
10,75 kN/m 2
3612,0 kN
5183,64 kN
Loading in fire situation:
GA .Gk 1,1 .Qk ,1 2 ,i .Qk ,i Ad
Nfi ,Sd 1,0.1562 ,4 0 ,7.2049 ,6 2997 ,12 kN
Nfi ,Sd Nfi ,Rd 2997,12 kN Nfi ,Rd
!
Worked Example 3: Composite columns 487
Section characteristics: 300
Steel grade:
S235
Concrete grade: C20/25 Steel section: Standard fire resistance:
4 ď Ś 25
HEB 600 R90
600
50
Nb,Rd,z = 7973,9 kN 50
Worked Example 3: Composite columns 488
Restrictions on the application of Annex F:
l 13,5b = 13,5.300 = 4050 mm 230 mm h 1100 mm 230 mm b 500 mm S 235 steel grade S 460 C 20/25 concrete C 50/60 1 reinforcement 6 % R 120 min For R90-120: min (h, b) = 300mm For R90 - 120: h/b > 3 l 10 b
l= 2000 mm h = 600 mm b = 300 mm S 235 C 20/25 1,3 R = 90 min b = 300 mm h/b = 2 < 3
Worked Example 3: Composite columns 489
Flanges of the steel profile
• the average temperature
A f ,t 0 ,t kt m V 0,t = 805°C and kt = 6,15 300
600
Am 2 (h b ) 2 (0 ,3 0 ,6 ) 10 m 1 V h.b 0 ,3.0 ,6
Worked Example 3: Composite columns 490
The average temperature:
f ,t 0 ,t
Am kt V
f,t 805 6,15 (10 ) 866,5 C Yield strength:
kmax, 866 0,0767
Modulus of elasticity:
kE,866 0,0751
fa ,max,f , 235 .0 ,0767 18 ,04 MPa
E a ,f ,t 210000 .0 ,0751 15 ,781 .10 3 MPa
Worked Example 3: Composite columns 491
Design plastic resistance to axial compression in fire:
Nfi ,pl .Rd,f 2(b.ef .fa,max, f , ) / M,fi ,a
Nfi ,pl .Rd,f 2(300 .30.18,04 ) / 0,9 360,8 kN
Effective flexural stiffness in fire:
(
)
(EI )fi ,f ,z Ea,f ,t ef b3 / 6 15781,5.30.300 3 / 6 2130,5 kNm 2
Worked Example 3: Composite columns 492
Web of the steel profile: h w,fi
Neglected part hw,fi:
H hw ,fi 0 ,5 (h 2 ef )1 1 0 ,16 t h R90 Ht 1100 hw,fi
43 mm
Maximum stress level :
fa max,w ,t fay ,20 ,w
Ht 1 0 ,16 h
197,55 MPa
e f =30
Worked Example 3: Composite columns 493
Design plastic resistance to axial compression:
Nfi ,pl .Rd ,w ew ( h 2 ef 2 hw ,fi ).fa ,max,w , / M ,fi ,a Nfi ,pl .Rd ,w 15 ,5 (600 60 2.43 ).197 ,55 / 0 ,9 1544 ,6 kN
Effective flexural stiffness in fire:
( EI )fi ,w ,z Ea ,w ,20 (h 2 ef 2 hw ,fi )ew3 / 12 210000 .454 .15 ,5 3 / 12 20 ,59 kNm 2
Worked Example 3: Composite columns 494
Concrete
Neglected exterior layer bc,fi for R90:
bc,fi
Am 0,5 22,5 27,5 mm V
Concrete temperature for R90:
bc,fi
c,t = 357°C
Secant modulus of concrete for temperature 357°C: bc,fi
Ec. sec fc, / cu, fc,20 .kc, / cu, 2313,64MPa
Worked Example 3: Composite columns 495
Design plastic resistance to axial compression:
Nfi ,pl .Rd ,c 0,86(h 2.ef 2bc,fi )(b ew 2bc,fi ) As .0,85.fc,20 .kc, / M ,fi ,c 0,86(600 2.30 2.27,5)(300 15,5 2.27,5) 1964 .0,85.20.0,793 / 1 1267,7kN
Effective flexural stiffness in fire:
( EI )fi ,c ,z Ec ,sec, (h 2.ef 2 bc ,fi )(( b 2 bc ,fi )3 ew3 ) / 12 Is ,z
Is ,z
As .100 2.2 1964 .100 2 19 ,64.10 6 mm 2
( EI )fi ,c ,z 2313 ,64 (600 60 2.27 ,5 )(( 300 2.27 ,5 )3 15 ,5 3 ) / 12 4137 ,7 kNm 2
Worked Example 3: Composite columns 496
Reinforcement
(u
u1.u2 50.50 50mm
)
ky,t = 0,572
R90
kE,t = 0,406
A s=1964mm 2
50
u = 50 mm
4 25
Plastic resistance to axial compression :
Nfi ,pl .Rd,s As .k y ,t .fsy ,20 / M,fi ,s 365,11kN Effective flexural stiffness:
(EI )fi ,s,z kE,t .Es,20 .Is,z 1674,5 kNm 2
50
Worked Example 3: Composite columns 497
Plastic resistance to axial compression: 43
Nfi,pl.Rd = Nfi,pl.f + Nfi,pl.w + Nfi,pl.c + Nfi,pl.s
27,5
Nfi,pl.Rd = 360,8+1544,6+1267,7+365,11=3538,2 kN
Effective flexural stiffness:
For R90 : f , 0 ,8 ; w , 1,0 ; c , 0 ,8 ; s , 0 ,8
27,5
(EI)fi,eff,z = f, (EI)fi,f,z + w, (EI)fi,w,z + c, (EI)fi,c,z + s, (EI)fi,s,z
(EI)fi,eff,z = 0,8.2130,5+1,0.29,58+0,8.4137,73+0,8.1674,5 = 6388,77 kNm2 Euler critical buckling load:
Nfi ,cr ,z 2.(EI )fi ,eff ,z / (l )2 2 .6383,77 / (2)2 15751,3 kN
Worked Example 3: Composite columns 498
Non-dimensional slenderness :
Nfi ,pl ,R 0 ,461 Nfi ,cr ,z
Nfi ,pl ,R Nfi ,pl .Rd when M,fi,i 1,0 3347 ,7 kN
Design axial buckling resistance :
Nfi ,Rd ,z z .Nfi ,pl .Rd 3060 ,5 kN
(
c z
0 ,865 )
Check of the column:
Nfi ,Rd ,z Nfi ,Sd 3060 ,5 kN 2997 ,12 kN The column satisfies the conditions for R90 fire resistance
!